text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors continue their study of Hitchin systems, started in [\textit{A. Chervov} and \textit{D. Talalaev} [Theoret. Math. Phys. 140, 1043--1072 (2004)]. They consider curves that can be obtained from a projective line by gluing two subschemes and they describe the related moduli space of vector bundles, the dualizing sheaf, and module endomorphisms. Finally they show that these constructions lead to Hitchin systems and prove their integrability. The exposition is followed by numerous examples. Hitchin systems; gluing subschemes; integrable systems; singular algebraic curves; \(r\)-matrix Vector bundles on curves and their moduli, Relationships between algebraic curves and integrable systems, Singularities of curves, local rings, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Hitchin systems on singular curves. II: Gluing subschemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We discuss how the classification of finite simple groups is used and mention some specific applications to various other fields of mathematics as well as to group theory. finite groups; finite simple groups; applications of simple groups; Brauer groups; Riemann surfaces; polynomials; function fields Guralnick, Robert, Applications of the classification of finite simple groups.Proceedings of the International Congress of Mathematicians---Seoul 2014. Vol. II, 163-177, (2014), Kyung Moon Sa, Seoul Finite simple groups and their classification, Primitive groups, Coverings of curves, fundamental group, Algebraic field extensions Applications of the classification of finite simple groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0538.00016.]
The main result is the following theorem. Let \(\ell^{\nu}\) be a prime power and X be a regular Noetherian separated scheme of finite Krull dimension. Suppose \(\ell\) is invertible in \({\mathcal O}_ X\) and that all residue fields of X have bounded étale cohomological dimension for \(\ell\)-torsion sheaves. If \(\ell =2\) or 3, assume that X contains enough roots of unity. There is a strongly converging spectral sequence
\[
E_ 2^{p,q}=H^ p_{et}(X,{\mathbb{Z}}/\ell^{\nu}(i))\quad for\quad q=2i,\quad E_ 2^{p,q}=0\quad for\quad other\quad q
\]
converging to \((K/\ell^{\nu})_{q-p}(X)[\beta^{-1}]\) with differentials of bidegree (r,r-1). Here \([\beta^{-1}]\) is a localization by inversion of the Bott element \(\beta\). This has a lot of consequences. In particularly for affine curves X over a finite field we have
\[
| \# K_{2i- 2}(X)[\beta^{-1}]^ 1_{\ell}/\# K_{2i-1}(X)[\beta^{-1}]^ 1_{\ell}|_{\ell}=| \zeta (X,1-i)|_{\ell},\quad i\geq 2,
\]
for \(\ell\)-adique valuations \(| \quad |_{\ell}.\) Here \(K_*(X)[\beta^{-1}]^ 1_{\ell}\) is the homotopy inverse limit of the system \((K/\ell^{\nu})_*(X)[\beta^{-1}]\). There is also the corresponding result for the spectrum of integers of a number field and new Gysin sequences of \({\mathbb{Q}}^ 1_{\ell}\)-cohomologies. Lichtenbaum-Quillen conjecture; K-theory with coefficients; Atiyah- Hirzebruch spectral sequence; values of zeta-functions R. Thomason, The Lichtenbaum-Quillen conjecture for K/l[{\(\beta\)}-1], Proc. 1981 Conference at Univ. Western Ontario, to appear. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Étale and other Grothendieck topologies and (co)homologies The Lichtenbaum-Quillen conjecture for \(K/\ell_*[\beta^{-1}]\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let K be an algebraic function field in one variable over an algebraically closed field k of characteristic \(p>0\). Let L/K be a finite p-extension. The relationship between the Hasse-Witt invariants of L and K is given by the Deuring-Shafarevich formula. Shafarevich established the formula when L/K is unramified. He then used it to show that the Galois group of the maximal unramified p-extension of K is a free pro-p- group on \(\lambda\) generators, \(\lambda\) being the Hasse-Witt invariant of K.
In this paper we generalize Shafarevich's technique. Let S be a finite set of primes of K. For \(P\in S\) choose a finite p-extension \~K\({}_ P/K_ P\) with Galois group \(G_ P\). Let \(K_ T\) be the composite of all finite p-extensions L/K unramified outside of S and such that the local extension at any prime Q of L dividing a \(P\in S\) is isomorphic to a subfield of \~K\({}_ P/K_ P\). We use the Deuring-Shafarevich formula to show that the Galois group of \(K_ T/K\) is the p-profinite completion of the free product of the \(G_ P's\) and a free group on \(\lambda\) generators. algebraic function field; Hasse-Witt invariants; Deuring-Shafarevich formula; Galois group; maximal unramified p-extension; p-profinite completion Arithmetic theory of algebraic function fields, Ramification and extension theory, Galois theory, Algebraic functions and function fields in algebraic geometry The Deuring-Šafarevič formula revisited | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics. toric geometry; Fano varieties; weak Fano varieties; building sets; nested sets; graph associahedra; reflexive polytopes; graph cubeahedra; root systems Fano varieties, Toric varieties, Newton polyhedra, Okounkov bodies Notes on toric Fano varieties associated to building sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we prove (using reducible curves and an inductive procedure) the following results.
Theorem 1 (The maximal rank conjecture in \(\mathbb P^3)\): If \(d\geq (3g+12)/4\), \(g\geq 0\), there is a smooth connected curve \(C\subset \mathbb P^3\) of degree \(d\), genus \(g\), with general moduli and of maximal rank (i.e. for all \(k\) the restriction maps \(H^0(\mathbb P^3,\mathcal O_{\mathbb P^3}(k))\to H^0(C,\mathcal O_C(k))\) have maximal rank).
Theorem 2: There is a function \(u_3\) with \(\lim_{g\to \infty}u_3(g)= \frac12\) such that for every \(d\geq gu_3(g)\), there is a generally smooth component \(W(d,g;3)\) of \(\text{Hilb}(\mathbb P^3)\) with the right number of moduli and such that a general element of \(W(d,g;3)\) has maximal rank.
Later we extend Theorem 2 to \(\mathbb P^n\), \(n>3\), with \(\lim_{g\to \infty}u_n(g)=(n-2)/(n-1)\).
[For the entire collection see Zbl 0614.00006.] deformation; space curve; postulation; Hilbert scheme; moduli of curves; maximal rank conjecture E. Ballico and Ph. Ellia, Beyond the maximal rank conjecture for curves in \(\mathbf P^3\) , in Space curves , Lecture Notes in Math., vol. 1266, Springer-Verlag, Berlin, 1987, pp. 1-23. Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Beyond the maximal rank conjecture for curves in \(\mathbb P^3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0741.00079.]
One investigates relations between tau functions, dressing kernels, wave functions, and spectral asymptotics for KdV, KP, and AKNS situations in a determinant context where emphasis is on the continuous spectrum. dressing kernels; wave functions; spectral asymptotics Carroll, R.,Inverse Scattering and Applications, Contemp. Math. Vol. 122, Amer. Math. Soc., Providence, RI, 1991, pp. 23-28;NEEDS' 90, Springer-Verlag, New York, 1991, pp. 2-5. KdV equations (Korteweg-de Vries equations), Scattering theory for PDEs, Inverse problems for PDEs, Theta functions and abelian varieties, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) On the determinant theme for tau functions, Grassmannians, and inverse scattering | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ``This introductory chapter is a rambling through basic notions and results of local complex analysis based on local function theory, local algebra and sheaves. In focus are coherent analytic sheaves. We discuss four fundamental results: Coherence of structure sheaves in \S7, Finite mapping theorems in \S8, Coherence of ideal sheaves in \S9, Coherence of normalization sheaves in \S14. All local function theory originates from the Weierstrass Preparation and Division Theorems. These theorems form the cornerstones of \S1. In sections 2 to 6 we introduce and discuss basic notions. Dimension theory is developed in \S10, while \S11 is devoted to homological codimension, Cohen-Macaulay spaces, Noether property and analytic spectra. Pure dimensional reduced complex spaces look locally like analytically branched coverings of domains in \(\mathbb{C}^ n\). In \S12 we study such coverings. Sections 13 to 15 are dealing with normal spaces and (semi- )normalizations''. Cohen-Macaulay space; analytic spectrum; homological codimension; meromorphic function; normalization sheaf; Weierstrass algebra; local complex analysis; coherent analytic sheaves [28] R. Remmert, Local theory of complex spaces, Several complex variables, VII, Encyclopaedia Math. Sci. 74, Springer, Berlin, 1994, p. 7-96 &MR 13 | &Zbl 0808. Analytic spaces, Local analytic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Local theory of complex spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In [Can. Math. Bull. 34, 181-185 (1991; Zbl 0695.14014)] \textit{D. Clark} shows that the function \(f: \mathbb{Z}^*\to \mathbb{N}\), counting the number of ways the integer \(n\) can be factored as \(n= d_ 1 d_ 2\) with \(d_ 1+ d_ 2=\)square, is unbounded. The author repeats Clark's argument, which uses the infinite structure of the group of rational points of the elliptic curve given by \(y^ 2= x^ 3+ 2x\). Furthermore, he relates Clark's result to the unsolved problem of the existence of elliptic curves \(E_ n/ \mathbb{Q}\), given by the equation \(y^ 2= x^ 3+ nx\), of unbounded rank, via the function
\[
\bar f(n)= \#\{d \bmod \mathbb{Q}^{*_ 2}\mid d\text{ divides } n \text{ and } d+n/d= \text{square}\}.
\]
arithmetic function; number of factorizations of an integer; group of rational points; elliptic curves Arithmetic functions; related numbers; inversion formulas, Cubic and quartic Diophantine equations, Elliptic curves Some arithmetic functions associated with diophantine equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Naor-Reingold sequences with elliptic curves are used in cryptography due to their nice construction and good theoretical properties. Here the authors give a new bound on the linear complexity of these sequences thereby improving the previous one obtained by \textit{I. E. Shparlinski} and \textit{J. H. Silverman} [Des. Codes Cryptography 24, No. 3, 279--289 (2001; Zbl 1077.11504)] and holding in more cases. The following theorem is the main result:
For \(0<\delta <1\) and \(n<\tfrac 43 \log l+6\), the linear complexity \(L_{\mathbf a}\) of the sequence \((u_k)_{k=0}^{2^n-1}\) satisfies
\[
L_{\mathbf a}\geq 2^{n/2} l^{-2/3-\delta}/8
\]
for all but at most \(O((l-1)^{n-\delta}\) vectors \(\mathbf a\in(\mathbb F_l^*)^n\). The implied constant is absolute. Naor-Reingold pseudo-random function; linear complexity; elliptic curves Cruz, M.; Gómez, D.; Sadornil, D., On the linear complexity of the Naor-Reingold sequence with elliptic curves, Finite Fields Appl., 16, 329-333, (2010) Pseudo-random numbers; Monte Carlo methods, Applications to coding theory and cryptography of arithmetic geometry, Curves over finite and local fields, Cryptography On the linear complexity of the Naor-Reingold sequence with elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The book is a selfcontained introduction to some of the combinatorical techniques for dealing with multigraded polynomial rings, semi group rings, and determinantal rings. An important role play combinatorically defined ideals and their quotients with the aim to compute numerical invariants and resolutions using gradings more refined than the standard grading.
The book is subdivided in three parts: Monomial Ideals, Toric Algebras, and Determinants. It has altogether 18 chapters containing homological invariants of monomial ideals and their polyhedral resolutions, toric varieties, local cohomology, Hilbert schemes among other subjects, to show how the tools developed can be used for studying algebraic varieties with group actions. Each chapter begins with an overview and ends with notes and references. The book assumes the knowledge of commutative algebra (graded rings, free resolutions, Gröbner bases) and a little simplicial topology and polyhedral geometry. It is interesting for a wide audience of students and researchers.
The book may serve as a basis for a full year course on this topic. It contains a lot of exercises and hints for further studies. monomial ideal; toric algebra; Hilbert scheme; local cohomology; multigraded polynomial rings E. Miller and B. Sturmfels, \textit{Combinatorial commutative algebra}, Graduate Texts in Mathematics volume 227, Springer, Germany (2005). Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects and applications of commutative rings, Toric varieties, Newton polyhedra, Okounkov bodies, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Parametrization (Chow and Hilbert schemes), Linkage, complete intersections and determinantal ideals Combinatorial commutative algebra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K_0(\mathcal V_{\mathbb C})\) be the Grothendieck ring of all algebraic varieties over \(\mathbb C\). If \(X\) is a complex variety and \(n\geq 1\) is a natural number, denote by \(X^{(n)}\) the \(n\)-fold symmetric of \(X\). If \(\mu\colon K_0({\mathcal V}_{\mathbb C})\to A\) is an \(A\)-valued motivic measure (i.e. a ring homomorphism) then one can define the formal power series
\[
\zeta_{\mu}(X,t)=1+\sum_{n=1}^{\infty}\mu([X^{(n)}])t^n\in A[[t]],
\]
which is called the motivic zeta function of \(X\). \textit{M. Kapranov} [The elliptic curve in the \(S\)-duality theory and Eisenstein series for Kac-Moody groups, preprint, http://arxiv.org/math.AG/0001005] proved that if \(A\) is a field and \(X\) is a curve then \(\zeta_{\mu}(X,t)\) is a rational function. Moreover he asked whether this is a rational function for all varieties \(X\) of dimension \(\geq 2\) . In the paper under review the authors give a negative answer to Kapranov's question by proving the following.
Theorem: There is a field \(A\) and a motivic measure \(\mu\colon K_0({\mathcal V}_{\mathbb C})\to A\) such that for every smooth projective surface \(X\) with \(p_g(X)=h^{2,0}(X)\geq 2\), the motivic zeta function \(\zeta_{\mu}(X,t)\) is not rational. Grothendieck ring; motivic zeta function Larsen, M.; Lunts, V. A., \textit{motivic measures and stable birational geometry}, Mosc. Math. J., 3, 85-95, (2003) Rational and birational maps, Motivic cohomology; motivic homotopy theory Motivic measures and stable birational geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A complete grid is the projective closure of affine lines which are parallel to the coordinate axes and pass through a lattice of points. In the paper under review, the authors compute the generators and, in case of \(\mathbb{P}^3\), the Hilbert function of complete grids. They also study the Cohen-Macaulay and seminormality properties of the homogeneous coordinate ring of a (complete and, in the case of \(\mathbb{P}^3\), also incomplete) grid. lines, grids; Hilbert function Guida M., Orecchia F.: Algebraic properties of grids of projective lines. J. Pure Appl. Algebra 208, 603--615 (2007) Computational aspects of algebraic curves, Projective techniques in algebraic geometry Algebraic properties of grids of projective lines | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Siehe Abschn. VIII. Cap. 2, JFM 01.0170.01. Conics; Coordinate Systems Projective and enumerative algebraic geometry On Conics, Plane and Spherical, referred to Three Point Tangential Coordinates | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the review in Zbl 0717.11050. generalization of class field theory; local fields; Milnor K-group; integral projective scheme; Chow group; generalization of ramification theory; higher dimensional schemes; generalized Swan conductor; global fields Class field theory, Class field theory; \(p\)-adic formal groups, Higher symbols, Milnor \(K\)-theory, \(K\)-theory of global fields, \(K\)-theory of local fields, Ramification and extension theory, Formal groups, \(p\)-divisible groups, Generalized class field theory (\(K\)-theoretic aspects) Generalization of class field theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Inter-universal Teichmüller theory may be described as a sort of arithmetic version of Teichmüller theory that concerns a certain type of canonical deformation associated to an elliptic curve over a number field and a prime number \(l\leq 5\). We begin our survey of interuniversal Teichmüller theory with a review of the technical difficulties that arise in applying scheme-theoretic Hodge-Arakelov theory to diophantine geometry. It is precisely the goal of overcoming these technical difficulties that motivated the author to construct the nonscheme-theoretic deformations that form the content of inter-universal Teichmüller theory.
Next, we discuss generalities concerning ``Teichmüller-theoretic deformations'' of various familiar geometric and arithmetic objects which at first glance appear one-dimensional, but in fact have two underlying dimensions. We then proceed to discuss in some detail the various components of the log-theta-lattice, which forms the central stage for the various constructions of inter-universal Teichmüller theory. Many of these constructions may be understood to a certain extent by considering the analogy of these constructions with such classical results as Jacobi's identity for the theta function and the integral of the Gaussian distribution over the real line. We then discuss the ``inter-universal'' aspects of the theory, which lead naturally to the introduction of anabelian techniques.
Finally, we summarize the main abstract theoretic and diophantine consequences of inter-universal Teichmüller theory, which include a verification of the ABC/Szpiro conjecture. elliptic curve; number field; thety function; hyperbolic curve; anabelian geometry; ABC conjecture; Szpiro conjecture 12. S. Mochizuki, Inter-universal Teichmüller theory I-IV (2015); http://www.kurims. kyoto-u.ac.jp/motizuki/papers-english.html. Arithmetic ground fields for curves, Coverings of curves, fundamental group A panoramic overview of inter-universal Teichmüller theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study weight one specializations of the Euler system of Beilinson-Flach elements introduced by Kings, Loeffler and Zerbes [\textit{G. Kings} et al., Camb. J. Math. 5, No. 1, 1--122 (2017; Zbl 1428.11103)], with a view towards a conjecture of Darmon, Lauder and Rotger relating logarithms of units in suitable number fields to special values of the Hida-Rankin \(p\)-adic \(L\)-function. We show that the latter conjecture follows from expected properties of Beilinson-Flach elements and prove the analogue of the main theorem of \textit{F. Castellà} and \textit{M.-L. Hsieh} [``On the non-vanishing of generalized Kato classes for elliptic curves of rank 2'', Preprint, \url{arXiv:1809.09066}] about generalized Kato classes. Beilinson-Flach; Stark units; iterated integrals; Hida-Rankin \(p\)-adic \(L\)-function Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Beilinson-Flach elements, Stark units and \(p\)-adic iterated integrals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(H_0\) be a discrete periodic Schrödinger operator on \(\ell^2(\mathbb{Z}^d)\):
\[
H_0=-\Delta +V,
\]
where \(\Delta\) is the discrete Laplacian and \(V:\mathbb{Z}^d\rightarrow \mathbb{C}\) is periodic. We prove that for any \(d\ge 3\), the Fermi variety at every energy level is irreducible (modulo periodicity). For \(d=2\), we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for \(d=2\) and a constant potential \(V\), the Fermi variety at \(V\)-level has exactly two irreducible components (modulo periodicity). We also prove that the Bloch variety is irreducible (modulo periodicity) for any \(d\ge 2\). As applications, we prove that when \(V\) is a real-valued periodic function, the level set of any extrema of any spectral band functions, spectral band edges in particular, has dimension at most \(d-2\) for any \(d\ge 3\), and finite cardinality for \(d=2\). We also show that \(H=-\Delta +V+v\) does not have any embedded eigenvalues provided that \(v\) decays super-exponentially analytic variety; algebraic variety; Fermi variety; Bloch variety; irreducibility; extrema; band function; band edge; embedded eigenvalue; unique continuation; Landis' conjecture; periodic Schrödinger operator Schrödinger operator, Schrödinger equation, Projective techniques in algebraic geometry, Analytic subsets of affine space, Eigenvalue problems for linear operators, Families, moduli of curves (algebraic) Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is a review of analytic methods (\(L^2\) Hodge theory) used in algebraic geometry for studying adjoint linear systems, vanishing theorem for algebraic vector bundles and invariance of plurigenera of general type families. Among the topics discussed in the paper are singular metrics, applications to Fujita's conjecture [\textit{T. Fujita} in: Algebraic Geometry, Proc. Symp., Sendai 1985, Adv. Stud. Pure Math. 10, 167--178 (1987; Zbl 0659.14002)] on global generation of adjoint linear systems, and analytic tools in Siu's proof [\textit{Y.-T. Siu}, Invent. Math. 134, No. 3, 661--673 (1998; Zbl 0955.32017)] of invariance of plurigenera for a family of general type. adjoint linear systems; line bundle; Fujita conjecture; variety of general type; invariance of plurigenera; Hodge theory Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Transcendental methods of algebraic geometry (complex-analytic aspects), Adjunction problems \(L^2\) methods and effective results in algebraic geometry. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By the result of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], there exists an equivalence between the derived category of \(G-\)equivariant coherent sheaves on a quasiprojective variety \(M\) with the derived category of coherent sheaves on the irreducible component \(Y\) of the \(G-\)Hilbert scheme of \(M\) that contains free orbits. The equivalence holds under the assumption that \(G\) is a finite group acting on \(M\) such that the canonical bundle on \(M\) is locally trivial as a \(G-\)sheaf and \(\text{dim} Y\times_{(M/G)} Y \leq \dim M +1.\) The article under review generalizes this result to the case of any smooth Deligne-Mumford stack \(\mathcal X\) with coarse moduli space \(X\) which is a quasiprojective Gorenstein variety. Denote by \(\text{Hilb}({\mathcal X})\) a scheme representing the Hilbert functor studied by \textit{M. Olsson} and \textit{J. Starr} [Commun. Algebra 31, No. 8, 4069--4096 (2003; Zbl 1071.14002)]. Then, the role of scheme \(Y\) is played by the component \(\text{Hilb}'({\mathcal X})\subset \text{Hilb}({\mathcal X})\) containing non-stacky points in \(\mathcal X.\)
In the above setting, the main theorem asserts that if \(\text{dim} \text{Hilb}'({\mathcal X})\times_X \text{Hilb}'({\mathcal X}) \leq \dim X +1,\) then \(\text{Hilb}'({\mathcal X})\) is smooth and there is an equivalence between the categories \(D^b({\mathcal X})\) and \(D^b(\text{Hilb}'({\mathcal X}))\) given by the integral functor with universal object over \(\text{Hilb}'({\mathcal X})\) as a kernel.
Moreover, in the setting of Bridgeland, King and Reid, the authors prove the twisted version of equivalence in the sense of \textit{V. Baranovsky} and \textit{T. Petrov} [Adv. Math. 209, No. 2, 547--560 (2007; Zbl 1113.14033)]. McKay correspondence; derived categories; stacks; Hilbert scheme; Brauer group Chen, J-C; Tseng, H-H, A note on derived mckay correspondence, Math. Res. Lett., 15, 435-445, (2008) McKay correspondence, Stacks and moduli problems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), Brauer groups of schemes A note on derived McKay correspondence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper, of which three proofs are given (two of them due to L. Moret-Bailly and N. M. Katz, respectively) is the following theorem: Suppose that \(X\) is a separated scheme of finite type over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(G\) be a finite \(p\)-group acting freely on \(X\) and \(Y=X/G\). Then \(\chi_ c(X,{\mathbb Q}_ p)=| G| \cdot \chi_ c(Y,{\mathbb Q}_ p).\) Here \(\chi_ c\) denote the Euler-Poincaré characteristic for cohomology with compact supports.
There are some consequences of which the following three should be mentioned:
(1) the formula of Deuring-Shafarevich for the \(p\)-ranks of \(X\) and \(Y\) for a finite morphism \(f: X\to Y\) of smooth projective curves over \(k\) [cf. e.g. \textit{M. L. Madan}, Manuscr. Math. 23, 91--102 (1977; Zbl 0369.12011)];
(2) the fact that \(\pi_ 1(X)\) has no p-torsion [in the case of the algebraic closure of a finite field this was proved by \textit{T. Katsura}, C. R. Acad. Sci., Paris, Sér. A 288, 45--47 (1979; Zbl 0429.14011)]; and
(3) if \(p=2\) and \(X\) is a singular Enriques surface, then its double-cover \(K3\)-surface is ordinary. étale \(p\)-covers; torsionless fundamental group; group acting on scheme; \(p\)-ranks of smooth projective curves; characteristic \(p\); Euler-Poincaré characteristic; singular Enriques surface Crew, Richard M., Etale \(p\)-covers in characteristic \(p\), Compositio Math., 52, 1, 31-45, (1984) Coverings in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, Homotopy theory and fundamental groups in algebraic geometry, Finite ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients) Étale \(p\)-covers in characteristic \(p\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors describe methods for the computation of the quasiadjunction polytopes for plane curves singularities. These polytopes give refinements of the zero sets of multivariable Alexander polynomials.
As a consequence, some hyperplanes are determined on which all the polynomials in the multivariable Bernstein ideal vanish. plane curve singularities; Alexander invariants; local systems; mixed Hodge structure [12] Pierrette Cassou-Noguès &aAnatoly Libgober, &Multivariable Hodge theoretical invariants of germs of plane curves&#xJ. Knot Theory Ramifications20 (2011) no. 6, p. 787Article | &MR 28 | &Zbl 1226. Mixed Hodge theory of singular varieties (complex-analytic aspects), Singularities in algebraic geometry Multivariable Hodge theoretical invariants of germs of plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The subject of this note is the Vassiliev-Goodwillie-Weiss spectral sequence. This objects calculates an approximation of the homology of the space of knots. It has been proved by \textit{P. Lambrechts} et al. in [Geom. Topol. 14, No. 4, 2151--2187 (2010; Zbl 1222.57020)] that this spectral sequence degenerates at the \(E_2\)-page when working with rational coefficients. Besides, \textit{I. Volić} in [Compos. Math. 142, No. 1, 222--250 (2006; Zbl 1094.57017)] has shown that the knot invariant given by the \(E_2\)-page of this spectral sequence is the universal finite type invariant (or more precisely the associated graded one given by the filtration according to the grade). It has been conjectured that these two results remain true when working with integer coefficients.
In this note we are interested specifically in the first conjecture (to know whether the spectral sequence degenerates with integer coefficients). In joint work with Pedro Boavida de Brito we have introduced a new tool to study this problem. We construct a non-trivial action of the absolute Galois group of \(\mathbb Q\) on this spectral sequence. This action gives us information on the differentials. In this way we can show that a given differential is zero if we ignore the torsion for the first small integers (see Theorem 5.2 for the precise result).
\dots
Let us give some details on the contents of this note. The first section gives a quick introduction to the Goodwillie-Weiss embedding calculus. The second is a specialization of this theory to the case of embeddings from \(\mathbb R\) to \(\mathbb R^d\) following the work of \textit{D. P. Sinha} [J. Am. Math. Soc. 19, No. 2, 461--486 (2006; Zbl 1112.57004)]. The third section is an introduction to the Grothendieck-Teichmüller group and to the absolute Galois group of \(\mathbb Q\) and to their actions on the profinite completions of the pure braid groups. In the fourth section we recall the theory of completion of spaces at a prime number. This is an analog in homotopy theory of the completion of groups. Finally in the fifth section, we can give the main results together with a sketch of the proof. Note that only the results of this final part are original. They are due to Pedro Boavida de Brito and the author. A complete proof will be published soon. Vassiliev-Goodwillie-Weiss spectral sequence; space of knots; \(E_2\)-term; universal finite type invariant; Vassiliev invariant; integer coefficients; action; Galois group; differentials; completion Finite-type and quantum invariants, topological quantum field theories (TQFT), Spectral sequences in algebraic topology, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Galois group and knot space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We extend, to linear systems of forms of \(\mathbb P_r(\mathbb C)\) that have order \(n>2\) and indeterminate Jacobian, some results known for \(n=2\) and describe linear systems of quadrics of maximal dimension with preassigned dimension of Jacobian variety. linear systems of forms; indeterminate Jacobian; Jacobian with irregular dimension Divisors, linear systems, invertible sheaves, Hypersurfaces and algebraic geometry Linear systems of forms whose Jacobian has irregular dimension | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves an identity involving modular forms conjectured by \textit{S. Hosono, M.-H. Saitō} and \textit{J. Stienstra} [Integrable systems and algebraic geometry, Kobe/Kyoto, 1997, World Sci. Publishing, River Edge, NJ, 194--235 (1998; Zbl 0963.14016)]. Let \(E_8\) denote the standard even unimodular lattice of rank 8 and let \(\Theta_{E_8}\) denote its usual theta function. Then define the weight 4 modular form \(G(z)\) by \( G(z)=\frac{1}{9}q^{4/3}\Theta_{E_8}(3z,z\gamma)\) where \(\gamma\) is an element of \(E_8\) with norm 8. Define the power series \(\psi(u)\) and \(q(u)\) by
\[
\begin{aligned} \psi(u)&:= \exp \left (\frac{\mathcal{F}_1(u)}{\mathcal{F}(u)}\right ),\\ q(u)&:= u\exp \left ( 3\frac{\mathcal{F}_1(u)-\mathcal{F}_2(u)}{\mathcal{F}(u)} \right ), \end{aligned}
\]
where \(\mathcal{F}, \mathcal{F}_1\), and \(\mathcal{F}_2\) are hypergeometric in the sense that
\[
\begin{aligned} &\mathcal{F}(u)= F\bigg(\frac 13, \frac 23; 1; 27u\bigg),\\ &\mathcal{F}_1(u)=\sum_{n\geq 0}\frac{(3n)!}{n!^3}\left (1+\frac{1}{2}+\cdots +\frac{1}{3n}\right )u^n,\\ &\mathcal{F}_2(u)=\sum_{n\geq 0}\frac{(3n)!}{n!^3}\left ( 1+\frac{1}{2}+\cdots +\frac{1}{n}\right ) u^n. \end{aligned}
\]
If integers \(\beta(n)\) are defined by the expression
\[
\frac{u}{\psi(u)}\frac{q'(u)}{q(u)}=\sum_{n\geq 0}\beta(n)q(u)^n,
\]
then the author conjectures the following identity
\[
\sum_{n\geq 0}\beta(n)q^{n-{1}/{6}}=\frac{G(z)}{\eta^{12}(3z)}= \eta^{-4}(z)
\]
where \(\eta(z)\) denotes the Dedekind eta function. Dedekind eta-function, theta function Zagier, D; Saito, M-H (ed.); Shimizu, Y (ed.); Ueno, K (ed.), A modular identity arising from mirror symmetry, 477-480, (1998), Singapore Relations with algebraic geometry and topology, Holomorphic modular forms of integral weight, Theta series; Weil representation; theta correspondences, Calabi-Yau manifolds (algebro-geometric aspects) A modular identity arising from mirror symmetry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a compact connected Lie group and let \(T\) be a maximal torus of \(G\). In this paper the authors calculate the real \(KO\)-theory of a flag manifold \(G/T\) for the exceptional Lie groups \(G=G_2, F_4, E_6\) using the Atiyah-Hirzebruch spectral sequence and Steenrod squaring cohomology operations. Also they point out the connection between Witt groups and the real \(KO\)-theory of homogeneous spaces such as Grassmannians and flag manifolds. All calculations are done explicitly. real \(K\)-theory; exceptional Lie groups; flag manifolds; Atiyah-Hirzebruch spectral sequence; Witt groups; generalized cohomology Daisuke Kishimoto and Akihiro Ohsita, \?\?-theory of exceptional flag manifolds, Kyoto J. Math. 53 (2013), no. 3, 673 -- 692. Topological \(K\)-theory, Generalized cohomology and spectral sequences in algebraic topology, Grassmannians, Schubert varieties, flag manifolds \(KO\)-theory of exceptional flag manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is the first half of the author's work devoted to the systematic treatment of canonical spin polynomial invariants of algebraic surfaces. The notion of a spin polynomial of a smooth simple connected compact complex algebraic surface \(S\) was introduced in the author's recent works [see, e.g., Russ. Acad. Sci., Izv., Math. 42, No. 2, 333-369 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 2, 125-164 (1993; Zbl 0823.14031)] in the differential-geometric setting. In the present paper the spin polynomials are defined in the algebro-geometric setting. The paper consists of two chapters.
In chapter 1 a treatment of Jacobians \(J_k^i(S)\) and theta-loci \(\Theta_k^i(S)\) is given. Here \(i\) indexes the spin chambers of the Kähler cone of \(S\), \(J_k^i(S)\) is the Gieseker-Maruyama moduli space \(M^i(2, c_1=K_S, c_2=c_2(S)+k)\) with respect to the almost-canonical polarization lying in the \(i\)-th chamber, and \(\Theta_k^i(S)= \{[F]\in J_k^i(s)\mid h^0(F)\geq 1\}\). Then the spin polynomials \(s\gamma(k,n,i)\in S^dH^2(S,\mathbb{Z})\), \(d=3c_2(S)- K_S^2+3k-p_a(S)- 2n+1\), \(n\geq 0\), are defined by the cohomological correspondence using the discriminant of the universal quasifamily of sheaves on \(S\times \Theta_k^i(S)\). -- In chapter 2 the differential-geometric construction of spin polynomials is discussed and its coincidence with the algebro-geometric one is proved; the spin polynomials \(s\gamma(k,n,i)\) are then interpreted as ``fundamental'' algebraic classes of intermediate dimension \(2d\) in the cohomology ring of the Hilbert scheme \(\text{Hilb}^dS\). Jacobian; canonical spin polynomial invariants; theta-loci; Gieseker-Maruyama moduli space; almost-canonical polarization; Hilbert scheme Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Differentiable structures in differential topology, Moduli, classification: analytic theory; relations with modular forms, Parametrization (Chow and Hilbert schemes), Theta functions and abelian varieties Canonical spin polynomials of an algebraic surface. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let C be a smooth algebraic curve over a finite field of characteristic p\(>0\), and let A denote the Dedekind ring of all functions on C which are regular outside a fixed rational point \(\infty\). The author had introduced a new type of L-functions [Invent. Math. 55, 107-116; 117-119 (1979; Zbl 0402.12006 and Zbl 0402.12007)] with values in the \(\infty\)- adic completion K of the function field k of A and more general in finite extensions over K. These L-functions are defined by some sort of Dirichlet series for \(s=(s_ 0,s_ 1)\in K^{\times}\times {\mathbb{Z}}_ p\) and for \(\bar K-\)valued characters \(\chi\) on a certain extension of the 1-units \(U_ 1\) in K by the ideal class group of A in close analogy to the number field case. On the ''halfplane'' degree \((s_ 0)>0\) the L- series become obviously convergent and even have an Euler product. In fact they turn out to be ''entire''. A simplified version of the author's results on the special values at ''positive integer powers'' \(s=(\pi^{- i},i)\) (with \(\pi\) a uniformizer at \(\infty)\) for positive \(i\in {\mathbb{N}}\) is summarized [by the author in Duke Math. J. 47, 345-364 (1980; Zbl 0441.12002)] where remarkable similarities with cyclotomic fields show up, such as periods, Bernoulli-Carlitz ideals, von Staudt-Clausen type results and \(\Gamma\)-functions. Among other things there are algebraicity results for \(s_ 1=-i\), a negative integer, and just as in the cyclotomic number field case these values can be interpolated v-adically for any finite prime v of A. The interpretation of these facts in terms of distributions is discussed and a Kummer criterion for the p- divisibility of the class number is given. Most of the material in this article seems to have appared in the papers quoted above and in J. Algebra 81, 107-149 (1983; Zbl 0516.12010). cyclotomic function fields; algebraic curve over a finite field; \(L\)-functions Goss, D., On a new type of \textit{L}-functions for algebraic curves over finite fields, Pacific J. math., 105, 143-181, (1983) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry On a new type of \(L\)-function for algebraic curves over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors calculate the Betti numbers of the Hilbert scheme of points in the plane. Observe that the maximal torus of SL(3) acts on \(Hilb^ d({\mathbb{P}}^ 2)\) with isolated fixed points. It follows from a result of Birula-Białynicki that \(Hilb^ d({\mathbb{P}}^ 2)\) has a cellular decomposition. Then the calculation of the Betti numbers reduces to a careful study of the representation of the torus at the tangent spaces of the fixed points. As a by-product to their method, the authors also obtain similar results about the punctual Hilbert scheme and the Hilbert scheme of points in the affine plane. Betti numbers of the Hilbert scheme of points in the plane; punctual Hilbert scheme Ellingsrud, Geir; Strømme, Stein Arild, On the homology of the Hilbert scheme of points in the plane, Invent. Math., 87, 343-352, (1987) Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry On the homology of the Hilbert scheme of points in the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{B. Sturmfels} [J. Algebr. Comb. 3, No. 2, 207-236 (1994; Zbl 0798.05074)] has established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum of point configurations in \(\mathbb R^d\) and of coherent polyhedral subdivisions of the associated Cayley embedding -- the polyhedral version of the Cayley trick of elimination theory. The authors generalize this relationship in the case of non-coherent subdivisions. As an application, a new proof is given of the Bohne-Dress theorem on zonotopal tilings. polytopes; subdivision; zonotopal tiling Huber, Birkett; Rambau, Jörg; Santos, Francisco, The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings, J. Eur. Math. Soc. (JEMS), 2, 2, 179-198, (2000) \(n\)-dimensional polytopes, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(P_1, \dots, P_s\) be distinct points in \(\mathbb{P}^ n\), the projective \(n\)-space over an algebraically closed field \(K\), and let \(\wp_i\) be the homogeneous prime ideal in the ring \(R=K[X_0, \dots, X_n]\) defining \(P_i\) (\(i=1, \dots, s\)). Set \(I=\wp_1^{m_1} \cap \dots \cap \wp_s^{m_s}\), where \(m_i\) is a positive integer for every \(i\). The \(0\)-dimensional subscheme \(Z:=m_1P_1+ \dots +m_sP_s\) of \(\mathbb{P}^n\) defined by the homogeneous ideal \(I\) is called a set of fat points. The Hilbert function of \(Z\), denoted by \(H_Z\), is the Hilbert function of the associate coordinate ring \(R/I\), namely \(H_Z(t)=\dim_K((R/I)_t)\) for every \(t \in \mathbb{N}\). It is a strictly increasing function until it reaches the multiplicity of \(R/I\), \(\sum_{i=1}^s \binom{m_i+n-1}{n}\), at which it stabilizes. The regularity index \(\text{reg}(Z)\) of \(Z\) is the smallest integer \(t\) at which \(H_Z\) stabilizes. The calculation of \(\text{reg}(Z)\) is very difficult in general and has been obtained only in few particular cases, e.g. by \textit{E. D. Davis} and \textit{A. V. Geramita} [Curves Semin. at Queen's, Vol. 3, Kingston/Can. 1983, Queen's Pap. Pure Appl. Math. 67, Exp. H, 29 p. (1984; Zbl 0597.13014)] when the points are collinear. On the other hand, some upper bounds for \(\text{reg}(Z)\) are known. Let \(T_{jZ}\) be the maximum of the integers \( \lfloor \frac{1}{j}( \sum_{\ell=1}^q m_{i_{\ell}}+j-2)\rfloor\) for \(P_{i_1}, \dots P_{i_q}\) lying on a linear \(j\)-space, and set \(T_Z = \max\{T_{j Z} \vert j=1, \dots n \}\). By work of \textit{E. Ballico} et al. [J. Pure Appl. Algebra 220, No. 6, 2307--2323 (2016; Zbl 1332.13016)] and of \textit{U. Nagel} and \textit{B. Trok} [Segre's Regularity Bound for Fat Point Schemes, Arxiv 1611.06279 (2016)], it is known that \(\text{reg}(Z) \leq T_Z\) for any set \(Z\) of fat points. Equality holds in some instances, as shown by the first author in a series of papers, but not in general. In the paper under review, the authors show that \(T_Z -1 \leq \text{reg}(Z) \leq T_Z\) in the following cases: 1) \(Z\) is supported on two lines; 2) \(s \leq 5\); 3) \(s=n+3\) and \(P_1, \dots, P_{n+3}\) do not lie on a hyperplane of \(\mathbb{P}^ n\). fat points; regularity index; zero-scheme Divisors, linear systems, invertible sheaves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series An estimate of the regularity index of fat points in some cases | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f\) be a function germ of an analytic function at \(0\in \mathbb{R}^{n}.\) Then the classical Łojasiewicz inequality is
\[
\left\vert f(x)\right\vert \geq C\operatorname{dist}(x,f^{-1}(0))^{\alpha }
\]
in a neighbourhood of \(0\) (for some \(C,\alpha >0).\) The author reports on some recent results concerning analogs of this\ inequality and the best exponent \(\alpha \) (called the Łojasiewicz exponent) for smooth functions, both in local (as above) and global (in the whole \(\mathbb{R}^{n}\) for polynomials) cases. Most of these results are of his own or in cooperation.
Since for smooth functions the Łojasiewicz inequality does not hold in general, the results require additional assumptions on \(f\) or extension of the set \(f^{-1}(0).\) The author describes also some results concerning atypical values of polynomials. Łojasiewicz inequality; smooth function; atypical value Real algebraic and real-analytic geometry, Asymptotic behavior of solutions to equations on manifolds Smooth and global versions of the Łojasiewicz inequality | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a complex irreducible smooth projective curve, and let \(\mathbb{L}\) be an algebraic line bundle on \(X\) with a nonzero section \(\sigma_0\). Let \(\mathcal{M}\) denote the moduli space of stable Hitchin pairs \((E,\, \theta)\), where \(E\) is an algebraic vector bundle on \(X\) of fixed rank \(r\) and degree \(\delta\), and \(\theta \in H^0 (X,\, \mathcal{E}nd(E)\otimes K_X\otimes\mathbb{L})\). Associating to every stable Hitchin pair its spectral data, an isomorphism of \(\mathcal{M}\) with a moduli space \(\mathcal{P}\) of stable sheaves of pure dimension one on the total space of \(K_X\otimes\mathbb{L}\) is obtained. Both the moduli spaces \(\mathcal{P}\) and \(\mathcal{M}\) are equipped with algebraic Poisson structures, which are constructed using \(\sigma_0\). Here we prove that the above isomorphism between \(\mathcal{P}\) and \(\mathcal{M}\) preserves the Poisson structures. Hitchin pair; Poisson structure; moduli space Poisson manifolds; Poisson groupoids and algebroids, Symplectic structures of moduli spaces, Vector bundles on curves and their moduli, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Comparison of Poisson structures on moduli spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0676.00006.]
Consider the cup product map \(\mu\) : \(H^ 1(X,{\mathbb{Z}}/n)\times H^ 1(X,{\mathbb{Z}}/n)\to_ nB(X)\) for a connected scheme X over \({\mathbb{Z}}[1/n][\omega]\); \(\omega\) denotes a primitive n-th root of unity, \({}_ nB(X)\) is the subgroup of the Brauer group B(X) of X consisting of all elements of order dividing n and cohomology is taken with respect to the étale topology. If X is the spectrum of a field k this map is completely understood by the work of \textit{A. S. Merkur'ev} and \textit{A. A. Suslin} [see Sov. Math., Dokl. 25, 690-693 (1982); translation from Dokl. Akad. Nauk SSSR 264, 555-559 (1982; Zbl 0525.18007) and Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)]. Under certain conditions the author describes the image and the kernel of \(\mu\) of the spectrum of \(k[x_ 1,...,x_{\nu},f^{-1}]\) and for a fibre product space. cup product; Brauer group of a scheme Brauer groups of schemes, Varieties and morphisms, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) On the Brauer group and the cup product map | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathfrak O}\) be the ring of integers of a local field \(K\) of characteristic 0 with a residue field \(k\) of characteristic \(p>0\) and with ramification index \(e \leq p-1\); let \(\Gamma = \text{Gal} (\overline {K}/K)\). In this paper we obtain an explicit construction of finite commutative group schemes \(G\) over the ring \({\mathfrak O}\) that are annihilated by multiplication by \(p\); we obtain a description of them with the help of generalized finite Honda systems. We investigate the functor \(G \mapsto G (\overline K)\) (including the case \(e = p - 1)\), and we get sufficient (also necessary for \(e = 1)\) conditions under which a given \(\mathbb{F}_ p [\Gamma]\)-module can be realized as a \(\Gamma\)-module of \(\overline K\)-points \(G (\overline K)\) of the group scheme of \(G\) of the type considered. small ramification; characteristic \(p\); group schemes; generalized finite Honda systems Abrashkin, V.: Group schemes over a discrete valuation ring with small ramification, Leningrad Math. J. \textbf{1}(1), 57-97 Group schemes, Local ground fields in algebraic geometry, Galois theory, Valuation rings Group schemes over a discrete valuation ring with small ramification | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let C be a smooth, geometrically connected, projective algebraic curve defined over a field \(K\). Denote by \(J_ C\) the Jacobian and by \(F=K(C)\) the function field of \(C\) over \(K\), and let \(g=g_ F\) be the genus of F. To every separable, nonconstant morphism \(f: C\to C'\) of \(C\) to an algebraic curve \(C'\) over \(K\) with function field \(F'=K(C')\) of genus g'\(\geq 2\), one can associate an invariant \(\gamma^*_ E\) in the K- endomorphism ring \(End_ K(J_ C)\) of \(J_ C\) by taking
\[
\gamma^*_ E:=\gamma^*_ f:=f^*\circ f_*: J_ C\to^{f_*}J_{C'}\to^{f^*}J_ C\quad,
\]
where \(f_*\) (resp. \(f^*)\) is the direct (resp. inverse) image map acting on the respective Jacobian varieties and \(E\) is the image of \(F'\), \(E:=f^* F'\leq F\), a subfield of \(F\).
The classical theorem of de Franchis states that
a) the number of invariants \(\gamma^*_ E\in End_ K(J_ C)\) is finite, and
b) for any fixed invariant \(\gamma^*_ E\in End_ K(J_ C)\), there are at most finitely many subfields \(E/K\) of \(F/K\).
By modifying the classical proof of de Franchis in the setting of \textit{P. Samuel} [Lectures on old and new results on algebraic curves, Tata Inst. Fundam. Res. (Bombay 1966; Zbl 0165.241)], \textit{A. Howard} and \textit{A. J. Sommese} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 429-436 (1983; Zbl 0534.14016)], in the case of \(K={\mathbb{C}}\), gave an explicit upper bound for the number \(N_ F\) of non-rational subfields of \(F/K\) in terms of the genus g of \(F/K\). (An error in their result was corrected by the author.)
In the present very interesting and important paper, an ''Angle theorem'' is proved and combined with a ''packing argument'' due to Mumford to obtain a new and effective proof of de Franchis' theorem. In fact the author establishes the following refined explicit bound.
Let r be the rank of \(End_ K(J_ C)\). It is known that \(r\leq r(g)\) for \(r(g)=2g^ 2\) or \(=4g^ 2\), according as the characteristic of K is \(=0\) or \(\neq 0\), respectively. Then, if \(g>g'\geq 2\), the number \(N_ F(g')\) of separable subfields of \(F/K\) of genus \(g'\) satisfies the inequality
\[
N_ F(g')\leq (C+1)^{r-1} - (C-1)^{r-1},\text{ where } C:=2 \sqrt{\frac{(g-g')g'}{(g'-1)g}}\quad.
\]
It follows from this estimate that the function field F/K has at most
\[
(g-1) 2^{r(g)-2} (2^{r(g)-1} -1)
\]
separable subfields of genus \(\geq 2.\)
Slightly weaker versions of the main theorems of this paper had been announced earlier [Tagungsberichte, Math. Forschungsinstitut Oberwolfach 35/81, p. 11 (1981)]. number of non-rational subfields; number of separable subfields; number of morphisms of algebraic curves; Chow coordinates; theorem of the base; Jacobian; genus; function field; Angle theorem; de Franchis' theorem E. Kani, Bounds on the number of non-rational subfields of a function field, Invent. Math. 85 (1986), 185-198. Zbl0615.12017 MR842053 Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Birational geometry, Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves, Special surfaces Bounds on the number of non-rational subfields of a function field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the adjoint linear system \(|K_S + L|\) on a complex surface \(S\) of general type. For a nef divisor \(L\) on \(S\), we give a numerical criterion for \(\phi_{K_{S}+L}\) to be generically finite. A question of \textit{M. Chen} and \textit{E. Viehweg} [Pac. J. Math. 219, No. 1, 83--95 (2005; Zbl 1093.14056)] is studied and we give some partial evidence for this question to be true. algebraic surfaces; adjoint linear systems Surfaces of general type, Divisors, linear systems, invertible sheaves, Adjunction problems, Families, moduli, classification: algebraic theory Adjoint linear systems on algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a smooth irreducible complex projective surface \(X\), the Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) can be seen as a smooth resolution of the \(n\)-th symmetric product of \(X\). Many topological properties of \(X^{[n]}\) are known. The Betti numbers have been calculated by \textit{L. Göttsche} [Math. Ann. 286, No. 1--3, 193--207 (1990; Zbl 0679.14007)] and only depend on the Betti numbers of \(X\). This result has been clarified by \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] by constructing a representation of a Heisenberg algebra built from the rational cohomology of the surface on the direct sum \(\mathbb H := \bigoplus \mathrm H^*(X^{[n]}, \mathbb Q)\).
In the paper at hand, the author extends these results to Voisin's Hilbert schemes [\textit{C. Voisin}, Ann. Inst. Fourier 50, No. 2, 689--722 (2000; Zbl 0954.14002)] associated to compact almost-complex four-manifolds. He is able to prove both Göttsche's formula and the defining commutation relations of Nakajima's operators in this context. One main ingredient of the proof is Le Poitier's decomposition theorem for semi-small maps (following the decomposition theorem by \textit{A. A. Beilinson, J. Bernstein} and \textit{P. Deligne} [Faisceaux pervers. Astérisque 100, 172 p. (1982; Zbl 0536.14011)] without using any characteristic \(p\)-methods or étale cohomology), which is included together with a proof as it is otherwise unpublished.
Finally, tautological bundles are defined in this almost-complex setting. Hilbert scheme; Voison's Hilbert scheme; almost-complex four-manifolds; Göttsche formula; Nakajima operators J Grivaux, Topological properties of punctual Hilbert schemes of almost-complex fourfolds, to appear in Manuscripta Math. Almost complex manifolds, Parametrization (Chow and Hilbert schemes), \(4\)-folds Topological properties of Hilbert schemes of almost-complex fourfolds. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study motivic stable homotopy groups over finite base fields, specifically in weight zero, in a range. These computations are obtained in a number of steps: by the motivic Serre finiteness theorem, the groups \(\pi_{n,0}(F_q)\) are torsion. It follows that the \(l\)-primary part of these groups is given by the homotopy groups of the \(l\)-completion of the motivic sphere spectrum, which can be computed using the motivic Adams spectral sequence, as long as \(l \neq p\). In the range computed one finds that away from \(p\), \(\pi_{n,0}(F_q) = \pi_{n} \oplus \pi_{n+1}\). Here the right hand side denotes the ordinary stable homotopy groups. The authors also show that this pattern does not continue for \(n>18\).
The occurrence of the classical stable stems may at first appear surprising. The authors show further that if \(\bar{F}\) is an algebraically closed field of characteristic \(p\), then away from \(p\) we have \(\pi_{n,0}(\bar{F}) = \pi_n\), via the canonical map. This is an analogue of a theorem of Levine (who establishes this result for \(\bar{F}=\mathbb{C}\)). The result is proved by relating the completed motivic stable homotopy groups over a Henselian dvr (of mixed characteristic!), its residue and fraction field, using Spitzweck's theory of motivic cohomology. In the opinion of the reviewer, this is the most innovative part of the paper. motivic Adams spectral sequence; stable motivic stems over finite fields; computer-assisted motivic Ext group calculations Wilson, G. M.; Østvær, P. A., Two-complete stable motivic stems over finite fields, Algebr. Geom. Topol., 17, 2, 1059-1104, (2017) Motivic cohomology; motivic homotopy theory, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Adams spectral sequences, Applications of methods of algebraic \(K\)-theory in algebraic geometry Two-complete stable motivic stems over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present article studies birational anabelian reconstruction for function fields $K|k$ of varieties of dimension 1 over algebraically closed fields. This is an extension of the birational anabelian program initiated by \textit{F. A. Bogomolov} [in: Algebraic geometry and analytic geometry. Proceedings of a conference, held in Tokyo, Japan, August 13-17, 1990. Tokyo etc.: Springer-Verlag. 26--52 (1991; Zbl 0789.14021)] which aims at recovering function fields $K|k$ of dimension $> 1$ over algebraically closed fields from their absolute Galois group $G_K$. This cannot be possible in the one-dimensional case since then $G_K$ is profinite free of rank $|k|$ by results of \textit{D. Harbater} [Contemp. Math. 186, 353--369 (1995; Zbl 0858.14013)] and \textit{F. Pop} [Invent. Math. 120, No. 3, 555--578 (1995; Zbl 0842.14017)], containing therefore almost no information about $K$. The authors show however that $K|k$ can be recovered if, in addition to $G_K$, also the larger automorphism group $\mathrm{Aut}(K|k)\supseteq G_K$ fixing only the base field is provided. Also, there is found a Galois-type correspondence for transcendental field extensions and a group-theoretic characterisation of stabiliser subgroups for $\mathrm{PGL}(2,k)$ acting on $\mathbb P^1$. birational anabelian; algebraically closed fields; absolute Galois group; function fields; Galois-type correspondence Elliptic curves over global fields, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Algebraic functions and function fields in algebraic geometry A birational anabelian reconstruction theorem for curves over algebraically closed fields in arbitrary characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper adds to the research started recently by some of the authors and others concerning the so called \(\textit{unexpected}\) varieties. This concept correspond to the following:
Let \(Z\subset {\mathbb{P}^n}\) be a scheme and let \(X=\sum_i{m_i\lambda_i}\) where \(\lambda_i \subset \mathbb{P}^n\), \(1\le i \le r\), are general linear varieties of dimension \(\delta_i\) which are disjoint from \(Z\) and where each \(m_i\) is an integer. We say that \((n, X, Z, t)\) is \textit{unexpected} if vanishing on \(X\) imposes on \([I_Z]_t\) (forms of degree \(t\) in \(I_Z\)) fewer than the expected number of conditions.
The authors of this paper obtain three ways to produce unexpected varieties that expand on the knowledge on this type of varieties.
One way is the use of cones on positive dimensional varieties of codimension or more. It is proved that almost always give unexpected hypersurfaces.
For non cones, the authors improve previous results coming from finite sets of points and they construct unexpected surfaces coming from lines in \(\mathbb{P}^3\).
Finally, they generalize the construction, using birational transformations (Veneroni transformations) to obtain unexpected hypersurfaces in higher dimensions. cones; fat flats; special linear systems; line arrangements; unexpected varieties; base loci Configurations and arrangements of linear subspaces, Syzygies, resolutions, complexes and commutative rings, Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry, Combinatorial aspects of commutative algebra, Sheaves in algebraic geometry New constructions of unexpected hypersurfaces in \(\mathbb{P}^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author determines explicitly the Galois lattice structure of the Mordell-Weil group \(E_{\gamma}(\overline{\mathbb{Q}}(t))\) of all elliptic curves over the function field \(\overline{\mathbb{Q}}(t)\) with equation \(E_{\gamma}:\;y^2=x^3+\gamma x+t\) \((\gamma\in\overline{\mathbb{Q}}^{\times})\) with respect to the height pairing and the Galois group \(H\) of the field extension \(\mathbb{Q}(\zeta_{20})(\sqrt[20]{\gamma /G})/{\mathbb{Q}}(\gamma)\), \(G\) a well-determined real number in \(\mathbb{Q}(\zeta_{20})\). The lattice is isometric to the unique negative-definite quadratic even unimodular lattice \(E_8\) of rank 8. As \(H\)-module over \(\mathbb{Z}\) it is isomorphic to \(\mathbb{Z}[\zeta_{20}]\cdot \sqrt[20]{\gamma /G}\) of degree 8 of \(H\). Two applications to the corresponding Artin (respectively Hecke) \(L\)-function and Hasse zeta function are announced.
Computer calculations have been used. Details and more general accounts are promised to be published elsewhere. Galois lattice structure of the Mordell-Weil group; height pairing; L-function; Hasse zeta function; computer calculations Shioda, T.: The Galois Representations of TypeE 8 Arising from Certain Mordell-Weil Groups, Proc. Japan Acad.65A, 195--197 (1989) Rational points, Elliptic curves, Computational aspects of algebraic curves The Galois representation of type \(E_8\) arising from certain Mordell-Weil groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Elliptic curves find numerous applications. This paper describes a simple strategy to speed up their arithmetic in right-to-left methods. In certain settings, this leads to a non-negligible performance increase compared to the left-to-right counterparts. elliptic curve arithmetic; binary right-to-left exponentiation; mixed coordinate systems Joye, M.: Fast point multiplication on elliptic curves without precomputation. In: Gathen, J., Imaña, J.L., Koç, Ç.K. (eds.) WAIFI 2008. LNCS, vol. 5130, pp. 36--46. Springer, Heidelberg (2008) Cryptography, Finite ground fields in algebraic geometry, Computational aspects of algebraic curves, Elliptic curves Fast point multiplication on elliptic curves without precomputation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article gives a summary of the author's unpublished Ph.D thesis. It is known that Dynkin diagrams can be separated in two classes: the simply laced (or homogeneous) ones \(A_k\) (\(k\geq 1\)), \(D_k\) (\(k\geq 4\)), \(E_6\), \(E_7\) and \(E_8\), and the non-simply laced (or inhomogeneous) ones \(B_k\) (\(k\geq 2\)), \(C_k\) (\(k\geq 3\)), \(F_4\) and \(G_2\).
The aim of the article is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the homogeneous simple singularities to the inhomogeneous ones.
To a homogeneous simple singularity, one can associate the representation space of a particular quiver.
This space is endowed with an action of the symmetry group of the Dynkin diagram associated to the simple singularity which allows the construction and explicit computation of the semiuniversal deformations of the inhomogeneous simple singularities.
By quotienting such maps, deformations of other simple singularities are obtained.
In some cases, the discriminants of these last deformations are computed. simple singularities; quiver representations; root systems; foldings Deformations of singularities, Representations of quivers and partially ordered sets, Root systems Deformations of inhomogeneous simple singularities and quiver representations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We determine the zeta functions of trinomial curves in terms of Jacobi sums, and obtain an explicit formula of the genus of a trinomial curve over a finite field, and we study the conditions for this curve to be a maximal curve over a finite field. zeta function; trinomial curves; maximal curve; genus Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Curves over finite and local fields, Special algebraic curves and curves of low genus Zeta functions of trinomial curves and maximal curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Belyi's theorem and the associated theory of dessins d'enfants has recently played an important rôle in Galois theory, combinatorics and Riemann surfaces. In this mainly expository article we describe some consequences for Riemann surface theory. It is organised as follows: In \S 1 we describe the ideas of critical points and critical values which leads in \S 2 to the definition of a Belyi function. In \S 3 we state Belyi's theorem and define a Belyi surface. In \S 4 the close connection with triangle groups is described and this leads in \S 5 to an account of maps and hypermaps (or dessins d'enfants) on a surface. These are closely related to Belyi surfaces but in order to investigate this connection we introduce the idea of a smooth Belyi surface and a platonic surface in \S 6.
The only new result in this article is theorem 7.1 which describes the connection between regular maps and their underlying Riemann surfaces. Belyi function; Belyi's theorem; Belyi surface; dessins d'enfants; Riemann surfaces Singerman, D; Bujalance, E (ed.); Costa, AF (ed.); Martínez, E (ed.), Riemann surfaces, Belyi functions and hypermaps, 43-68, (2001), Cambridge Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Riemann surfaces, Belyi functions and hypermaps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We construct a double indexing family of reducible \(\text{SU} (2)\)- connections over \(S^ 2 \times S^ 2\) in a geometric way. By separation of variables, we compute the spectrum of the Hessian of the Yang-Mills functional at these connections. Using the implicit function theorem, we prove that these connections are isolated non-minimal solutions to the Yang-Mills equations on \(S^ 2 \times S^ 2\). 4-manifolds; \(\text{SU} (2)\)-connections; implicit function theorem; non- minimal solutions; Yang-Mills equations Variational problems concerning extremal problems in several variables; Yang-Mills functionals, Implicit function theorems; global Newton methods on manifolds, \(4\)-folds, Topology of Euclidean 4-space, 4-manifolds, Special Riemannian manifolds (Einstein, Sasakian, etc.) The construction of isolated reducible \(SU(2)\)-connections over \(S^ 2\times{}S^ 2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X = \text{Spec}(R)\) be a reduced equidimensional algebraic variety over an algebraically closed field \(k\). Let \(Y = \text{Spec}(R/{\mathfrak q})\) be a codimension one ordinary multiple subvariety, where \({\mathfrak q}\) is a prime ideal of height \(1\) of \(R\). If \(U\) is a nonempty open subset of \(Y\) and \({\mathfrak m}\) a closed point of \(U\), we denote by \(A\cong R_{\mathfrak m}\) its local ring in \(X\), by \({\mathfrak p}\) the extension of \({\mathfrak q}\) in \(A\), and by \(K\) the algebraic closure of the residue field \(k({\mathfrak p})\).
Then there exists a bijection \(\gamma^{\mathfrak m}:\text{Proj}(G_{\mathfrak p}(A) \otimes A/{\mathfrak p}) \to \text{Proj}(G(A_{\mathfrak p}) \otimes_{k({\mathfrak p})}K)\) such that for every subset \(\Sigma\) of \(\text{Proj}(G_{\mathfrak p}(A) \otimes_{A/{\mathfrak p}}k)\), the Hilbert function of \(\Sigma\) coincides with the Hilbert function of \(\gamma^{\mathfrak m}(\Sigma)\). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety. generic position; Hilbert function; tangent cone Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Families, moduli of curves (analytic) The structure of the tangent cone: an interesting bijection | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems classification of Brauer groups; rational function fields over global fields; Ulm invariants B. Fein, M.M. Schacher and J. Sonn, Brauer groups of rational function fields, Bull. Amer. Math. Soc. 1, 766-768. Arithmetic theory of algebraic function fields, Galois cohomology, Transcendental field extensions, Brauer groups of schemes Brauer groups of rational function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We use an Adams spectral sequence to calculate the \(\mathbb{R}\)-motivic stable homotopy groups after inverting \(\eta\). The first step is to apply a Bockstein spectral sequence in order to obtain \(h_1\)-inverted \(\mathbb{R}\)-motivic \(\mathrm{Ext}\) groups, which serve as the input to the \(\eta\)-inverted \(\mathbb{R}\)-motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor-Witt \((4k-1)\)-stem has order \(2^{u+1}\), where \(u\) is the 2-adic valuation of \(4k\). This answer is reminiscent of the classical image of \(J\). We also explore some of the Toda bracket structure of the \(\eta\)-inverted \(\mathbb{R}\)-motivic stable homotopy groups. motivic homotopy theory; stable homotopy group; eta-inverted stable homotopy group; Adams spectral sequence Guillou, B. J.; Isaksen, D. C., The \({\eta}\)-inverted \(\mathbb{R}\)-motivic sphere, Algebr. Geom. Topol., 16, 5, 3005-3027, (2016) Motivic cohomology; motivic homotopy theory, Adams spectral sequences, Stable homotopy of spheres The \(\eta\)-inverted \(\mathbb{R}\)-motivic sphere | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give geometric constructions of families of graded Gorenstein Artin algebras, some of which span a component of the space \(\text{Gor}(T)\) parametrizing Gorenstein Artin algebras with a given Hilbert function \(T\). This gives a lot of examples where \(\text{Gor}(T)\) is reducible. We also show that the Hilbert function of a codimension four Gorenstein Artin algebra can have an arbitrarily long constant part without having the weak Lefschetz property. graded Gorenstein Artin algebras; Hilbert function; codimension four Gorenstein Artin algebra DOI: 10.2140/pjm.1999.187.1 Cohen-Macaulay modules, Commutative Artinian rings and modules, finite-dimensional algebras, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Authors' abstract: It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification. discrete integrable systems; dynamical systems; Painlevé equations; algebraic geometry A.S. Carstea, A. Dzhamay and T. Takenawa, \textit{Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations}, \textit{J. Phys.}\textbf{A 50} (2017) 405202 [arXiv:1702.04907]. Automorphisms of surfaces and higher-dimensional varieties, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Discrete version of topics in analysis Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let F be a number field, A its ring of algebraic integers, and \({\mathfrak P}\) any maximal ideal of A. Denote by \(\hat A_{{\mathfrak P}}\) and \(\hat F_{{\mathfrak P}}\) the \({\mathfrak P}\)-adic completion of A and F, respectively. - Let \(f\in F[x,y]\); Igusa's local zeta function of f is defined by
\[
Z(s)=\int_{\hat A^ 2_{{\mathfrak P}}}| f(x,y)|^ s| dx\wedge dy|
\]
for \(s\in {\mathbb{C}}\), \(Re(s)>0\). A well-known list of candidate poles of (the meromorphic continuation to \({\mathbb{C}}\) of) Z(s) is given as follows.
Let \(\Pi\) : \(X\to \hat F^ 2_{{\mathfrak P}}\) be an embedded resolution of \(f=0\) and \(E_ j\), \(j\in T\), the irreducible components of \(\Pi^{- 1}(f^{-1}\{0\})\) with numerical data \((N_ j,\nu_ j)\). Then all real poles of Z(s) can be expressed as \(s=-\nu_ j/N_ j\), \(j\in T\). - Let \(s_ 0\in \{-\nu_ j/N_ h| j\in T\}\) not be induced by a component of the strict transform of \(f=0\). Then, for almost all \({\mathfrak P}\), we show: \(s_ 0\) is a pole of Z(s) if and only if at least one \(E_ j\) with \(s_ 0=-\nu_ j/N_ j\) occurs such that \(E_ j\) intersects the remaining components of \(\Pi^{-1}(f^{-1}\{0\})\) in at least 3 points. poles of Igusa's local zeta function; resolution of singularities Veys, W.: On the poles of Igusa's local zeta function for curves. J. Lond. Math. Soc. 41, 27--32 (1990) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta functions and \(L\)-functions On the poles of Igusa's local zeta function for curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author surveys numerous applications of the Siciak extremal function in so-called Constructive Function Theory. He supplements a comprehensive list of applications drawn up by \textit{M. Klimek} in his monograph [Pluripotential theory, Oxford etc.: Clarendon Press (1991; Zbl 0742.31001)]. Among the applications mentioned by the author are complex polynomial approximations, L-regularity of subsets of a complex affine space, invariance of L-regularity, Markov's inequality, Berstein-type theorems and algebraicity criteria for complex analytic sets.
The presentation is concentrated on problems that have been investigated by the author and his co-workers. Siciak extremal function; pluricomplex Green function; Berstein-Walsh-Siciak inequality; Julia sets; Markov and Berstein type inequalities Pleśniak, W., Siciak's extremal function in complex and real analysis, Ann. Polon. Math., 80, 37-46, (2003) Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs, Plurisubharmonic extremal functions, pluricomplex Green functions, Semi-analytic sets, subanalytic sets, and generalizations, Semialgebraic sets and related spaces, Real-analytic and semi-analytic sets, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets Siciak's extremal function in complex and real analysis | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0667.00008.]
The goal of this article is to understand better the structure of the Hilbert scheme \(H(p)=Hilb^ p({\mathbb{P}}^ n_ k)\) of projective subschemes of \({\mathbb{P}}^ n_ k\) with Hilbert polynomial p by considering linkage not just of individual subschemes but instead of entire flat families of them, in effect a study of linkage behavior under very general deformation.
For S locally noetherian, an S-point of D(p,q) is a sequence of closed embeddings XYP of flat S-schemes such that for any \(s\in S\) the schemes \(X_ s\) and \(Y_ s\) have Hilbert polynomials p and q, respectively. Further D(p;\textbf{f})\({}_{CM}\) is the open subscheme of D(p,q) for which the fibers \(X_ s\) are Cohen-Macaulay and equidimensional and \(Y_ s\) are complete intersections of multidegree \textbf{f}\(=f_ 1,...,f_ r\) for all \(s\in S\). The main result is that linkage of families X and \(X'\) with respect to a family of complete intersections Y defines an isomorphism D(p;\textbf{f})\({}_{CM}\to D(p';{\mathbf{f}})_{CM}\). If U is a subset of \(H(p)_{CM}\), all the members of U are contained in complete intersections of the same type, and \(U'\) is the set of linked subschemes in \(H(p')_{CM}\), then, under various additional hypotheses, properties of U (openness, irreducibility, smoothness of H(p) along it) can be carried over to \(U'\). As a corollary, if \(X\in H(p)_{CM}\) is non- obstructed, linked to \(X'\), and certain cohomological conditions hold on X and its ``generizations'', then \(X'\) is also non-obstructed. The author also gives a number of concrete examples, and methods for constructing these.
If, for example X is a (locally Cohen-Macaulay) curve in \({\mathbb{P}}^ 3\) with \(H^ 1(N_ X)=0\), then by linking geometrically by Y to \(X'\) (subject to certain constraints on the degrees for Y) and then again by suitable \(Y'\) to \(X''\), one obtains an obstructed curve that is doubly linked to X. Further results concern invariance of the cotangent sheaf \(A^ 2_ X\) and the obstruction space \(A^ 2(XY)\) under geometric linkage. There is also a subgroup C(XY) of \(A^ 2(XY)\) that is still invariant under appropriate conditions, but easier to compute and with useful applications to calculation of \(H^ 1(N_ X)\) for generic complete intersection curves in \({\mathbb{P}}^ 3\). For related results on invariance of cotangent modules the reader may consult \textit{R.-O. Buchweitz} [Thesis, Paris VII (1981)], \textit{R.-O. Buchweitz} and \textit{B. Ulrich} [``Homological properties invariant under linkage'' (preprint 1983)], \textit{B. Ulrich} [Math. Z. 196, 463-484 (1987; Zbl 0657.13023)], and for results on linkage of curves in \({\mathbb{P}}^ 3\), the sequence of papers by Bolondi and Migliore. liaison of families; obstructed curve; Hilbert scheme; Hilbert polynomial; linkage behavior under very general deformation; linkage of families Kleppe, J. O.: Liaison of families of subschemes in pn. Lecture notes in math. 1389 (1989) Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Formal methods and deformations in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Liaison of families of subschemes in \({\mathbb{P}}^ n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a henselian trait with a perfect residue field of characteristic \(p>0\). Let \(\eta\) and \(\tilde{\eta}\) be respectively the generic and geometric generic point of \(S\), \(G\) the Galois group of \(\tilde{\eta}\) over \(\eta\), \(X\) a scheme and \(f:X\rightarrow S\) a morphism of finite type. Let \(\ell \neq p\) be a prime number, \(\Lambda\) a finite field of characteristic \(\ell\) and \(\mathcal{K}\) a bounded below complex of étale sheaves of \(\Lambda\)-modules on \(X\). Introduced by Grothendieck in the 1960s, the nearby cycles complex \(R\Psi (\mathcal{K},f)\) and the vanishing cycles complex \(R\Phi (\mathcal{K},f)\) of \(\mathcal{K}\) relative to \(f\) are complexes on the geometric special fiber of \(f\) endowed with a \(G\)-action, called the monodromy action. The authors give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes' and Saito's logarithmic ramification filtration. This provides a positive answer to a conjecture of Leal for smooth morphisms in equal characteristic. The ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait is also studied. henselian trait; étale sheaf; abelian scheme; monodromy Étale and other Grothendieck topologies and (co)homologies, Ramification and extension theory, Structure of families (Picard-Lefschetz, monodromy, etc.) Characteristic cycle and wild ramification for nearby cycles of étale sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most \(r\). A natural algebraic representation of this problem gives rise to a determinantal ideal: the ideal generated by all minors of size \(r + 1\) of the matrix. We give new complexity bounds for solving this problem using Gröbner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0-dimensional and radical system of bi-degree \((D, 1)\). We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem. MinRank; Gröbner basis; determinantal; bi-homogeneous; structured algebraic systems J.-C. Faugère, M. Safey El Din, and P.-J. Spaenlehauer, \textit{On the complexity of the generalized MinRank problem}, J. Symbolic. Comput., 55 (2013), pp. 30--58. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Analysis of algorithms and problem complexity, Determinantal varieties, Vector spaces, linear dependence, rank, lineability, Cryptography On the complexity of the generalized MinRank problem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Es wird untersucht, welche Wertegruppen für eine Bewertung eines algebraischen Funktionenkörpers in \(n\) Veränderlichen in Frage kommen. Dabei wird von vornherein angenommen, daß alle Konstanten den Wert 0 haben und die Wertegruppe \(\Gamma\) eine Untergruppe der additiven Gruppe der reellen Zahlen ist. Der Rang einer solchen Gruppe \(\Gamma\) ist die Maximalzahl der rational linear unabhängigen Zahlen in \(\Gamma\). Nun wird bewiesen: Die Wertegruppe einer Bewertung \(V\) eines algebraischen Funktionenkörpers in \(n\) Veränderlichen hat entweder einen Rang \(r < n\) oder ist direkte Summe von \(n\) unendlichen zyklischen Gruppen. Ist umgekehrt eine Gruppe von reellen Zahlen \(\Gamma\) gegeben, die entweder einen Rang \(r < n\) hat oder direkte Summe von \(n\) zyklischen Gruppen ist, so gibt es eine Bewertung des rationalen Funktionenkörpers \(L = K(x_1, \dots, x_n)\) mit der Wertegruppe \(\Gamma\). Das wichtigste Hilfsmittel bei der Konstruktion einer solchen Bewertung ist die Existenz einer beliebigen Anzahl von algebraisch unabhängigen Potenzreihen in einer Veränderlichen über einem beliebigen Konstantenkörper. algebraic function fields; valuation; value group; rank; direct sum of n infinite cyclic groups MacLane, S. - Schilling, O.F.G.\(\,\): Zero-dimensional branches of rank 1 on algebraic varieties, Annals of Math. 40 (1939), 507-520 Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Valued fields Zero-dimensional branches of rank one on algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main aim of the paper is to generalize the notion of the Newton polytope of a polynomial to several more abstract contexts. The authors begin with proving general results on semigroups of integral points. Let \(S\) be an additive semigroup in the lattice \(\mathbb{Z}^n\subset\mathbb{R}^n\). Consider the group \(G(S)\) consisting of all the linear combinations \(\sum_i k_ia_i\), where \(a_i\in S\) and \(k_i\in\mathbb{Z}\), and the closed convex cone \(\text{Con}(S)\) that is the closure of the set of all linear combinations \(\sum_i \lambda_ia_i\) for \(a_i\in S\) and \(\lambda_i\geq 0\). Let us define the regularization of \(S\) as the semigroup \(\text{Reg}(S)=G(S)\cap\text{Con}\). Theorem~1.6 states that, for every closed strongly convex cone \(\text{Con}\) inside \(\text{Con}(S)\) that intersects the boundary of \(\text{Con}(S)\) only at the origin, there exists a constant \(N>0\) such that each point in the group \(G(S)\) that lies in \(\text{Con}\) and whose distance from the origin is bigger than \(N\) belongs to \(S\).
Assume that the cone \(\text{Con}(S)\) is strognly convex and denote by \(L\) its linear span in \(\mathbb{R}^n\). Let \(\dim L=q+1\). Fix a rational \(q\)-dimensional subspace \(M_0\subset L\) intersecting \(\text{Con}(S)\) only at the origin. Let \(M_k\), \(k\in\mathbb{Z}_{\geq 0}\), be the family of \(q\)-dimensional affine subspaces parallel to \(M_0\) such that each \(M_k\) intersects the cone \(\text{Con}(S)\) as well as the group \(G(S)\). The Hilbert function \(H_S(k)\) of a semigroup \(S\) is defined as the number of points in \(S\cap M_k\). Let us define the Newton-Okounkov body of \(S\) as the intersection \(\Delta(S)=\text{Con}(S)\cap M_1\). The authors prove that the function \(H_S(k)\) grows like \(a_qk^q\), where the coefficient \(a_q\) is equal to the (normalized in the appropriate way) \(q\)-dimensional volume of \(\Delta(S)\). Also the growth of the Hilbert function corresponding to the semigroup generated by \(S\cap M_k\) is characterized.
The second part is devoted to graded algebras and their Hilbert functions. Let \(F\) be a finitely generated extension of an algebraically closed field \(\mathbb{K}\) and \(F[t]\) be the algebra of polynomials over \(F\). Consider a nonzero finite dimensional subspace \(L\) of \(F\) over \(\mathbb{K}\) and let \(A_L:=\bigoplus_{k\geq 0}L^kt^k\). It is a homogeneous \(\mathbb{K}\)-subalgebra of \(F[t]\) generated by finitely many elements of degree \(1\). A homogeneous subalgebra \(A\) in \(F[t]\) is said to be of integral type if it is a finite module over some subalgebra \(A_L\). Moreover, a homogeneous subalgebra \(A\) is of almost integral type if it is contained in an algebra of integral type.
Every \(\mathbb{Z}^n\)-valued valuation of the field \(F\) maps the set of nonzero elements of a homogeneous subalgebra \(A\subseteq F[t]\) to a semigroup of integral points in \(\mathbb{Z}^n\times\mathbb{Z}_{\geq 0}\). This allows to define the Newton-Okounkov body of a graded algebra \(A\) and interpret the results on semigroups stated above in terms of graded algebras. In particular, it is proved that the Hilbert function of an algebra of almost integral type has polynomial growth.
The third part of the paper deals with algebraic geometry. Let \(X\) be an \(n\)-dimensional irreducible variety over \(\mathbb{K}\) with \(F=\mathbb{K}(X)\) being the field of rational functions. With a finite-dimensional subspace \(L\subset F\) one associates the Kodaira rational map \(\Phi_L: X \to \mathbb{P}(L^*)\). Let \(Y_L\) be the closure of the image of this map. The algebra \(A_L\) can be identified with the homogenenous coordinate ring of \(Y_L\subseteq\mathbb{P}(L^*)\). Algebras of integral type are related in these terms to the rings of sections of ample line bundles, while algebras of almost integral type are related to the rings of sections of arbitrary line bundles (see Theorems~3.7 and~3.8).
The Fujita approximation theorem in the theory of divisors states that the so-called volume of a big divisor can be approximated by the self-intersection numbers of ample divisors. The above results on graded algebras can be regarded as an abstract analogue of this result. In fact, it leads to a generalization of Fujita's result for any divisor or even any graded linear system on any complete variety (Corollary~3.11).
In the last part, a far-reaching generalization of the Kushnirenko theorem is obtained and a new version of the Hodge inequality is found. Also the authors give elementary proofs of the Alexandrov-Fenchel inequality and its analogue in algebraic geometry. semigroup of integral points; convex body; graded algebra; Hilbert function; Hodge index theory; mixed volume; Alexandrov-Fenchel inequality; Bernstein-Kushnirenko theorem; Cartier divisor; linear series Kaveh, K. and Khovanskii, A., Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, \textit{Ann. of Math. (2)}176 ( 2012), no. 2, 925- 978. Homogeneous spaces and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a commutative unital ring. A \(k\)-supermodule \( \mathfrak g =\mathfrak g_0\oplus \mathfrak g_1\) is a Lie superalgebra with bilinear multiplication \([x,y]\) such that standard skew-symmetry and Jacobi identities hold. Moreover \([w,w]= [z,[z,z]]=0\) for all even \(a\) and for all add \(z\). Also there exists a unary quadratic operation \(z^{(2)}\) mapping \(\mathfrak g_1\to \mathfrak g_0 \) such that
\[
(z_1+z_2)^{(2)} = z_1^{(2)} + z_2^{(2)} + [z_1,z_2],\quad [z^{(2)}x]=[z,[z,x]].
\]
For a affine supergroup \(G\) one can consider representing commutative Hopf \(k\)-superalgebra \(H=H_0\oplus H_1\). It is shown that in most cases the following property of being strongly split holds, namely, \(W^H=H_1/H_0^+H_1\) is \(k\)-free and there exists an isomorphism
\[
H\to (H/HH_1H)\otimes_k \left(\wedge W^H\right)
\]
of super counital \(\left (H/HH_1H\right)\)-comodule algebras. A Harish-Chandra pair over \(k\) consists of an affine \(k\)-group scheme \(G_+\) and a Lie superalgebra \(\mathfrak g\) where \(\mathfrak G_1\) is a free \(k\)-module of a finite rank. Moreover the Lie algebra of \( G_+\) is equal to \(\mathfrak g_0\) and there an action of \(G_+\) on \(\mathfrak g_0\) is extended to an action on \(\mathfrak g\). In a standard way with super Hopf algebra \(H\) one can associate super Lie algebra \(\text{Lie}(G)\). There exists a functor \(\Phi\) from the category of supergroups in which \(\mathfrak g_1\) for its Lie algebra is a free module of a finite rank to the category of Harish-Chandra pairs, namely \(\Phi(G)= (G_0, \text{Lie} G)\). One of the main results show that \(\Phi\) has an inverse functor and there is given an explicit for of this inverse. affine supergroup; Lie superalgebra; Hopf superalgebra; group scheme Gavarini F., Global splittings and super Harish-Chandra pairs for affine supergroups, Trans. Amer. Math. Soc. (2015), 10.1090/tran/6456. Lie (super)algebras associated with other structures (associative, Jordan, etc.), Group schemes, Hopf algebras and their applications Global splittings and super Harish-Chandra pairs for affine supergroups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a zero-dimensional subscheme \(\mathbb{X}\) of \(\mathbb{P}^2\), we bound the number of points in the support of \(\mathbb{X}\) which have maximal degree in \(\mathbb{X}\). For reduced schemes \(\mathbb{X}\) this yields a lower bound for the colength of the conductor \({\mathcal F}\) of the homogeneous coordinate ring \(R\) of \(\mathbb{X}\) in its integral closure \(\overline R\). This bound is attained by Castelnuovo sets for which we calculate \(\ell(R/{\mathcal F})\) explicitly.
Using the canonical decomposition of \(\mathbb{X}\), we also show a sharp upper bound for the colength \(\ell(R/{\mathcal F})\). Applications include estimates for the singularity degree \(\ell(\overline R/R)\) and the superabundance \(\ell(\overline R/R)-\ell(R/{\mathcal F})\) of \(\mathbb{X}\). Hilbert function; zero-dimensional subscheme; number of points; Castelnuovo sets DOI: 10.1016/S0022-4049(97)00164-3 Low codimension problems in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry Extremal zero-dimensional subschemes of \({\mathbb{P}}^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We compute the parameters of the linear codes that are associated with \textit{all} projective embeddings of Grassmann varieties. Grassmann codes; projective systems; parameters of a code Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory), Grassmannians, Schubert varieties, flag manifolds, Rational points Higher Grassmann codes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a solvable two-dimensional piecewise linear chaotic map that arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the non-trivial ultradiscrete limit of the solution in spite of a problem known as `the minus-sign problem.' ultradiscretization; tropical geometry; theta functions; plane cubic curve; discrete dynamical systems Kajiwara K., Kaneko M., Nobe A. and Tsuda T., Ultradiscretization of a solvable two-dimensional chaotic map associated with the hesse cubic curve, Kyushu J. Math. 63 (2009), no. 2, 315-338. Jacobians, Prym varieties, Elliptic curves, Theta functions and abelian varieties, Dynamical systems involving maps of the interval, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Strange attractors, chaotic dynamics of systems with hyperbolic behavior Ultradiscretization of a solvable two-dimensional chaotic map associated with the Hesse cubic curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) be an elliptic curve and let \(R\) be a real three dimensional root system. Let \(W\) be the Weyl group associated to \(R\), and put \(W_+=W\cap\text{SL}_3(\mathbb{C}).\) Then \(W_+\) acts naturally on \(E\otimes Q(R)\) and the quotient \(E\otimes Q(R)/W_+\) is singular with two natural crepant resolutions. One is the result of a Jung process of desingularization of singularities, the other the equivariant Hilbert scheme.
The author compares these resolutions case by case by writing up explicit equations, and in the Hilbert scheme case, an explicit example is given and a McKay correspondence is achieved. Finally, this correspondence results in a family of vector bundles on \(E\) parameterized by the \(W_+\)-Hilbert scheme. This article gives an interesting comparison of the two types of crepant resolutions in the dimension three case. crepant resolution; singularities; equivariant Hilbert scheme Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory, Local deformation theory, Artin approximation, etc. Resolution of non-abelian three-dimensional singularities. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper presents a lecture on existing algorithms for solving polynomial systems with their complexity analysis from our experiments on the subject. It is based on our studies of the complexity of solving parametric polynomial systems. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper provides different methods and techniques used for representing solutions of algebraic systems that include Rational Univariate Representations (RUR), Gröbner bases, etc. symbolic computations; complexity analysis; algebraic polynomial systems; parametric systems; rational univariate representations; Gröbner bases; triangular sets; irreducible components Ayad, A.: A survey on the complexity of solving algebraic systems. Int. Math. Forum 5(7), 333--353 (2010) Solving polynomial systems; resultants, Number-theoretic algorithms; complexity, Symbolic computation and algebraic computation, Effectivity, complexity and computational aspects of algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A survey on the complexity of solving algebraic systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A geometric approach to immersion formulas for soliton surfaces is provided through new cohomologies on spaces of special types of \(\mathfrak{g}\)-valued differential forms. We introduce Poincaré-type lemmas for these cohomologies, which appropriately describe the integrability conditions of Lax pairs associated with systems of PDEs. Our methods clarify the structure and properties of the deformations and soliton surfaces for the aforesaid Lax pairs. Our findings allow for the generalization of the theory of soliton surfaces in Lie algebras to general soliton submanifolds. Techniques from the theory of infinite-dimensional jet manifolds and diffieties enable us to justify certain common assumptions of the theory of soliton surfaces. Theoretical results are illustrated through \(\mathbb{CP}^{N-1}\) sigma models. cohomology; \(\mathbb{CP}^{N-1}\) sigma model; generalized symmetries; \(\mathfrak{g}\)-valued differential forms; \(\mathfrak{g}\)-valued de Rham cohomology; integrable systems; immersion formulas; soliton surfaces KdV equations (Korteweg-de Vries equations), Surfaces in Euclidean and related spaces, Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems, de Rham cohomology and algebraic geometry, de Rham theory in global analysis, Solitary waves for incompressible inviscid fluids, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) A cohomological approach to immersed submanifolds via integrable systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the book under review is to present a very powerful approach based on modern methods of monodromy theory to the study of many problems from singularity theory, algebraic geometry and theory of differential equations.
In the preface the author outlines the historical background of development of the monodromy theory from B. Riemann till present time and briefly describes the contents of his book. He also underlines that essential parts of the book are based on a two-year course at Warsaw University given by the author more than ten years ago.
In the first four chapters the author gives an introduction to the notion of monodromy in applications to multi-valued holomorphic functions and their Riemann surfaces, to the Morse theory in the real domain and the theory of normal forms for functions, he also describes some basic facts from the algebraic topology of manifolds and fibre bundles as well as from the topology and monodromy of analytic and algebraic functions including the Milnor theorem, Picard-Lefschetz formula, root systems, Coxeter groups, resolution and normalization of singularities, etc.
The next chapter 5 is devoted to the study of integrals along vanishing cycles, including basic results from the theory of Gauss-Manin connection, Picard-Fuchs systems, oscillating integrals and their relations with singularity theory. Then a method of abelian integrals is explained in detail; this chapter contains a few good examples and description of results Khovanski, Gabrielov, Petrov, and others.
It should be remarked that most of the above materials can also be found in the book [\textit{V. I. Arnol'd}, et al., Singularities of differentiable maps. Volumes I, II. Monographs in Mathematics, Vol. 82, 83. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001, 1988; Zbl 0659.58002)] .
In chapter 7 general ideas of the theory of Hodge structures and period mappings are described. Starting from classical results on the Hodge structure on algebraic manifolds, the author goes into the theory of mixed Hodge structures on incomplete manifolds and on the cohomological Milnor bundle, he also discusses the notions of limit Hodge structures in the sense of Schmid and Steenbrink, relations with the monodromy theory, etc. In conclusion, some basic constructions due to D. Mumford and Ph. Griffiths from the theory of period mappings in algebraic geometry are considered [see, for example, \textit{P. A. Griffiths}, Bull. Am. Math. Soc. 76, 228--296 (1970; Zbl 0214.19802)].
The subject of the next chapter 8 are the non-autonomous linear differential systems \(\dot z = A(t)z\), \( z\in \mathbb{C}^m,\) and the linear higher order differential equations \(x^{(n)} + a_1(t)x^{(n-1)} + \cdots + a_n(t)x = 0, \, x\in \mathbb{C},\) where \(t\in {\mathbb C}\) and all the entries of the matrix \(A(t)\) as well as the coefficients \(a_i(t)\) are meromorphic functions. The author discusses the notion of regular and irregular singularities, some aspects of the global theory of linear differential equations, the Riemann-Hilbert problem with a detailed analysis of Bolibruch's counter-example, the notion of isomonodromic deformations and some applications and relations with quantum field theory.
The chapters 9 and 10 are mainly concentrated on the local and global theory of holomorphic foliations in \(\mathbb{C P}^2,\) respectively. They contain the theory of resolution of vector fields, the theory of resurgent functions in the sense of Ecalle, the theory of Martinet-Ramis modules, theorems of Bryuno and Yoccoz, some results concerning the nonlinear Riemann-Hilbert problem [\textit{P. M. Elizarov}, et al., Nonlinear Stokes phenomena. Providence, RI: American Mathematical Society. Adv. Sov. Math. 14, 57--105 (1993; Zbl 1010.32501)], Ziglin theory, and a lot of other very interesting material with a number of useful examples.
In chapter 11 the author presents the basic notions and tools of differential Galois theory including the theory of Picard-Vessiot extensions with applications to the problem of integration of polynomial vector fields (Singer's theorem). He then discusses the monodromy theory of algebraic functions including the topological proof of Abel-Ruffini theorem, Khovanski's generalization of the monodromy group for large class of functions, the monodromy properties of Singer's first integrals.
The concluding chapter 12 is mainly based on the famous books of \textit{F. Klein} [see `Vorlesungen über die hypergeometrische Funktion'. Berlin: Springer (1933; Zbl 0007.12202 and JFM 59.0375.11), and so on.] It contains classical results from the theory of hypergeometric functions and explicit calculations of the monodromy and solutions of hypergeometric equations in quadratures. In conclusion two kinds of generalizations of hypergeometric functions are described. The first one is based on the Picard-Deligne-Mostow approach while the second -- on the approach of I. Gelfand, A. Varchenko, and others.
The book under review is written in a very clear and concise style, almost all key topics are followed by carefully chosen examples, non-formal remarks and comments. It contains a large number of pictures which may be considered as real visualizations of the discussed ideas. The bibliography includes 359 selected references. This makes the exposition not only accessible to beginning graduate students but highly interesting and useful for advanced research workers in algebraic and differential topology, algebraic geometry, complex analysis, differential equations and related fields of pure and applied mathematics and mathematical physics. monodromy theory; isolated singularities; critical points; Picard-Lefschetz theory; Gauss-Manin connection; Picard-Fuchs systems; oscillating integrals; abelian integrals; Hodge structures; period mappings; regular and irregular singularities; Riemann-Hilbert problem; isomonodromic deformations; holomorphic foliations; differential Galois theory; hypergeometric functions \.Zoł\polhk adek, Henryk, The monodromy group, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)] 67, xii+580 pp., (2006), Birkhäuser Verlag, Basel Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Monodromy on manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.), Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to global analysis, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Period matrices, variation of Hodge structure; degenerations, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Deformations of complex singularities; vanishing cycles, Mixed Hodge theory of singular varieties (complex-analytic aspects), Milnor fibration; relations with knot theory, Critical points of functions and mappings on manifolds, Singularities of vector fields, topological aspects, Topological invariants on manifolds, Differential algebra The monodromy group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(d\in\mathbb{N}\). The main object of study in this paper are families of Fermat hypersurfaces
\[
X_\lambda: \sum^n_{i=0} x^{d_i}_i+ \lambda\prod^n_{i=0} x^{a_i}_i,\quad\lambda\in \mathbb{F}_q,
\]
defined over the finite field \(\mathbb{F}_q\) of \(q\) elements. Here \(d_i\) is a natural number and \(a_i\) is a nonnegative integer, satisfying the conditions: \(d_iw_i= d\) for \(0\leq i\leq n\) and \(\sum^n_{i=0} w_i a_i= d\), so that \(X_\lambda\) may be regarded as a subvariety of the weighted projective space \(\mathbb{P}(w_0,\dots, w_n)\); moreover, it is assumed that h.c.f. \((q, d)= 1\).
The main technical result of the paper implies that the \(p\)-adic Picard-Fuchs equation, associated with the family \(X_\lambda\), is a generalised hypergeometric differential equation. The author discusses the Frobenius action on the cohomology of Fermat hypersurfaces and proves some results on the structure of the zeta function of a monomial deformation of such a hypersurface. As a by-product of his investigations, the author extends and improves on some results of \textit{S. Kadir} and \textit{N. Yui} [Motives and mirror symmetry for Calabi-Yau orbifolds. Modular forms and string duality, Fields Institute Communications 54, 3--46 (2008; Zbl 1167.14023)]. Fermat hypersurface; \(p\)-adic Picard-Fuchs equation; Monsky-Washnitzer cohomology; monomial deformations; zeta function Kloosterman, Remke, The zeta function of monomial deformations of {F}ermat hypersurfaces, Algebra \& Number Theory, 1, 4, 421-450, (2007) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Varieties over finite and local fields, Hypersurfaces and algebraic geometry, Rigid analytic geometry, Arithmetic ground fields (finite, local, global) and families or fibrations The zeta function of monomial deformations of Fermat hypersurfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In 1976, P. Deligne introduced a Fourier transformation into algebraic geometry, i.e. an involution for \(\ell\)-adic sheaves on the affine line (more generally on a vector bundle over a scheme of finite type) over a field \(k\) of finite characteristic \(p\ne \ell\) (depending on a character \(\mathbb F_p\to\mathbb Q_{\ell})\). The properties of this Fourier transformation have been studied by \textit{N. M. Katz} and the author [Publ. Math., Inst. Hautes Étud. Sci. 62, 145--202 (1985; Zbl 0603.14015)].
In the present paper a local version of Fourier transformation is introduced, defined on \(\ell\)-adic representations of the Galois group of a local field of equal characteristic. At the end of the paper, these Fourier transformations are used to give a new proof of the main result of \textit{P. Deligne}'s paper [Publ. Math., Inst. Haut. Étud. Sci. 52, 137--252 (1980; Zbl 0456.14014)] which in turn implies the part of the Weil conjectures which is the analog of the Riemann hypothesis.
The main part of the paper is devoted to Grothendieck's \(L\)-function for a complex \(K\) of \(\ell\)-adic sheaves on a curve \(X\) over \(k\). This \(L\)-function satisfies a functional equation for \(t\to t^{-1}\) in which a constant \(\varepsilon(X,K)\) appears. The main result of the paper under review expresses this constant as a product of local constants in the places of the curve. The proof of this product formula reduces to the case \(X=\mathbb P^1\) and then uses the Fourier transformation to investigate a one parameter deformation of a certain complex.
The author also explains the relevance of his product formula for Langlands' conjectures on the correspondence between \(L\)-functions and automorphic representations. geometric Fourier transformation; involution for \(\ell \)-adic sheaves; Weil conjectures; Riemann hypothesis; L-function; Langlands' conjectures Laumon, G., Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil, Publ. Math. Inst. Hautes Études Sci., 65, 131-210, (1987) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Langlands-Weil conjectures, nonabelian class field theory, Arithmetic problems in algebraic geometry; Diophantine geometry Fourier transformat, constants of the functional equations and Weil conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For all integers \(r\ge 3\), \(d\ge r\), \(g\ge 0\) Let \(\mathcal{H}^{L}_{d,g,r}\) denote the union of the irreducible component Hilbert scheme of \(\mathbb {P}^r\) whose general element is a smooth, connected and linearly normal curve \(C\) with degree \(d\) and genus \(g\) (over an algebraically closed field with characteristic zero). The paper always works in the Brill-Noether range \(\rho(d,g,r) =g-(r+1)(g-d+r)\ge 0\). Since \(C\) is linearly normal, \(h^1(C,\mathcal{O}_C(1)) =g-d+r\). In this range the wide-open Modified Assertion of Severi asks if \(\mathcal{H}^{L}_{d,g,r}\). The author had previously solved the cases \(g-d+r\le 3\). In the paper under review the author studies the case \(g-d+r=4\). He proves the existence part (with a few exceptions) and the irreducibility part for \(3\le r\le 8\). For the latter part he construct explicit families on certain surfaces. Hilbert scheme; algebraic curves; linearly normal; special linear series Parametrization (Chow and Hilbert schemes), Plane and space curves On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) with small index of speciality | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We survey some recent results on curves over finite fields with many rational points. zeta-function Curves over finite and local fields, Rational points, Arithmetic theory of algebraic function fields Algebraic curves over finite fields with many rational points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author deals with the following problem. Given a set \(Z\) of \(m\) points in \(\mathbb{P}^ n\) \((n \geq 2)\) and an integer \(r \geq 1\), is there a nondegenerate meromorphic vector field of degree \(r\) on \(\mathbb{P}^ n\) (i.e. a section \(s \in H^ 0 (\mathbb{P}^ n, T \mathbb{P}^ n(r))\) which has isolated zeroes, each of multiplicity one; \(T\mathbb{P}^ n=\) tangent bundle of \(\mathbb{P}^ n)\) which vanishes on \(Z\) and what is the dimension of the space of such vector fields?
Let \(\omega (n,r) = [(h^ 0 (\mathbb{P}^ n,T \mathbb{P}^ n (r)) - 1) /n]\), \({\mathcal I}_ Z=\) ideal sheaf of \(Z\). The author proves that for a general \(Z\) one has
(a) for \(m>n\), \(h^ 0(\mathbb{P}^ n, T \mathbb{P}^ n (r) \bigotimes {\mathcal I}_ Z) = \max (0,h^ 0 (\mathbb{P}^ n, T \mathbb{P}^ n (r)) - nm)\); in particular the former is nonzero if and only if \(m \leq \omega (n,r)\).
(b) If \(m \leq \omega (n,r)\) then a general \(s \in H^ 0 (\mathbb{P}^ n, T \mathbb{P}(r) \bigotimes {\mathcal I}_ Z)\) is nondegenerate.
The second main result is the following: Assume the base field has characteristic 0. Let \(Z\) be a 0-dimensional subscheme of \(\mathbb{P}^ n\). If \(h^ 1 (\mathbb{P}^ n, {\mathcal I}_ Z (r)) = 0\) then
\[
h^ 0 (\mathbb{P}^ n, T \mathbb{P}^ n (r) \otimes {\mathcal I}_ Z) = h^ 0 (\mathbb{P}^ n, T \mathbb{P}^ n (r)) - n \text{ (length} Z)
\]
and a general \(s \in H^ 0 (\mathbb{P}^ n, T \mathbb{P}^ n (r) \otimes {\mathcal I}_ Z)\) has zero scheme \((s)_ 0=\) union of \(Z\) and \((c_ n (T \mathbb{P}^ n(r)) - \text{length} Z)\) reduced points. In particular a general \(s\) is nondegenerate if \(Z\) is reduced. zero scheme of meromorphic vector field Ballico, E.: On meromorphic vector fields on projective spaces. Am. J. Math. 115(5), 1135-1138 (1993) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Projective techniques in algebraic geometry On meromorphic vector fields on projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Any projective module \(L\) over a noetherian ring \(A\) has the cancellation property, i.e. if \(L\oplus P\simeq L'\oplus P\) for some finitely generated projective \(A\)-module \(P\), then \(L\simeq L'\). The usual proof of this fact uses exterior powers. Here an elementary proof of this fact is given, which moreover outlines an explicit isomorphism \(\varphi:L\to L'\). It generalizes to noetherian schemes. noetherian scheme; projective module; cancellation property Projective and free modules and ideals in commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Cancellation of line bundles: Elementary proof | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the current paper, the authors continue their systematic study of the birational geometry of projective varieties of maximal Albanese dimension. Their general approach of characterizing birational properties of complex irregular varieties in terms of certain cohomological support loci on their Albanese varieties has been successfully developed in a series of foregoing papers [\textit{J. A. Chen} and \textit{C. D. Hacon}, Math. Ann. 320, No. 2, 367--380 (2001; Zbl 1073.14507); \textit{J. A. Chen} and \textit{C. D. Hacon}, Duke Math. J. 111, No. 1, 159-175 (2002; Zbl 1055.14010); \textit{C. D. Hacon} and \textit{R. Pardini}, J. Reine Angew. Math. 546, 177--199 (2002; Zbl 0993.14005)], and it is pursued further in their present won with regard to Iitaka fibrations.
More precisely, let \(L\) be a Cartier divisor on a projective variety \(X\) with Kodaira dimension \(\text{kod}(X,L)\geq 0\). The Iitaka fibration associated to \(L\) is a birational model of the map induced by the linear series \(|mL|\) for \(m\) being sufficiently big and divisible, which will be denoted by \(f_L: X'\to Y_L\), where \(X'\) stands for an appropriate birational model of \(X\). Being a fundamental tool in the birational classification of higher-dimensional varieties, the geometric properties of the image of the Iitaka fibration are of particular interest and importance. In this vein, the authors show how to characterize the irregularity of the variety \(Y_L\) in terms of the cohomological support loci
\[
V_m(L):=\{P\in \text{Pic}^0(X)\mid h^0(X, L^{\otimes m}\otimes P)\neq 0\}\text{ and }V:= \bigcup_{m\geq 1} V_m.
\]
One of their main results is that the maximal irreducible component of the variety \(V\) passing through the origin is in fact the subgroup \(f^*\text{Pic}^0(Y_L)\) of \(\text{Pic}^0(X)\), and that the irregularity of the Iitaka image \(Y_L\) equals, \(\dim(f^*\text{Pic}^0(Y_L))\).
Although the locus \(V\) itself need not be a group in general, the authors nevertheless show that the special case of the canonical divisor, that is, the case where \(L= K_X\), is very particular in that \(V\) is then really a subgroup of \(\text{Pic}^0(X)\) such that its components consist of finitely many torsion translates of \(f^*\text{Pic}^0(Y_{K_X})\). From this fact it is derived that, for all \(P\in V\), the corresponding Iitaka fibration \(f_{K_X\otimes P}\) is birationally equivalent to the canonical Iitaka fibration \(f_{K_X}\).
Finally, the authors study the case when the projective variety \(X\) is of maximal Albanese dimension. In this case, the locus \(T\) given by the intersection of all translates through the origin of the components of \(V_1(K_X)\) turns out to be of crucial importance for understanding the geometry of \(X\). In particular it is proved that if \(X\) is of general type and of maximal Albanese dimension, then \(X\) is of irregularity \(\dim(X)\), provided that the first plurigenus of \(X\) satisfies \(P_1(X)= 1\).
In this context the authors also give negative answers to related questions due to \textit{J. Kollár} [Shafarevich maps and automorphic forms, Princeton, NJ: Princeton University Press (1995; Zbl 0871.14015)]. fibrations; birational maps; Iitaka fibration; divisors; linear systems; Albanese variety; irregular algebraic varieties J.\ A. Chen and C.\ D. Hacon, On the irregularity of the image of the Iitaka fibration, Comm. Algebra 32 (2004), 203-215. Divisors, linear systems, invertible sheaves, Subvarieties of abelian varieties, Fibrations, degenerations in algebraic geometry, Rational and birational maps, Algebraic theory of abelian varieties On the irregularity of the image of the Iitaka fibration | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathfrak g}\) be a semisimple complex Lie algebra with a real form \({\mathfrak g}_{\mathbb R}\) and let \(G\), \(G_{\mathbb R}\) be the corresponding Lie groups. Assume that \(G_{\mathbb R}\) is connected and contains a compact maximal torus \(T\) with Lie algebra \({\mathfrak t}\). Let \({\mathfrak b}\) be a Borel subalgebra of \({\mathfrak g}\) containing a Cartan subalgebra which is the complexification of \({\mathfrak t}\) and let \(B\) be the corresponding Borel subgroup of \(G\). Let \(K_{\mathbb R}\) be the unique connected maximal compact subgroup of \(G_{\mathbb R}\) containing \(T\) and let \(K\) be the complexification of \(K_{\mathbb R}\). Write \(B_K:=K\cap B\), \(Z:=K/B_K\) and \(D:=G_{\mathbb R}/T\) (with an appropriate complex structure).
For any anti-dominant weight \(\mu\), there is an associated homogeneous line bundle \(L_\mu\) over \(D\). The Harish-Chandra module \(V^\mu\) can be realised as \(V^\mu:=H^d(D,L_\mu)\), where \(d\) is the dimension of \(Z\). The completion \(\widehat{V}^\mu\) is obtained by restricting \(H^d(D,L_\mu)\) to the formal neighbourhood of \(Z\) in \(D\). The modules \(V^\mu\) and \(\widehat{V}^\mu\) satisfy a linear PDE system. It is the object of this paper to investigate the differential relations satisfied by \(V^\mu\) and \(\widehat{V}^\mu\) in the context of the complex geometry of flag domains and the related correspondence and cycle spaces.
There are three filtrations of a certain cochain complex \((C^\bullet({\mathfrak n},\widehat{{\mathcal O}G}_{\mathcal W})_{-\mu},d)\); in fact, there are four filtrations in all, since the first has a variant which turns out to be more important. Each of these filtrations leads to a spectral sequence abutting to the completion \(\widehat{H}^q(D,L_\mu)\) of \(H^q(D,L_\mu)\) along \(Z\). The first filtration (and its variant) arise from a naïve consideration of the order of vanishing along the inverse image of \(Z\), the second gives rise to the Hochschild-Serre spectral sequence and the third is an amalgam of the other two. There are important compatibilities among the filtrations. Sections 3 to 6 of the paper are concerned with the filtrations, the associated spectral sequences and the relationships between them.
In Sections 7 and 8, the authors make the assumption that \(H^q(Z,\wedge^pN_{Z/D}^*\otimes L_\mu)=0\) for \(0\leq q\leq d-1\) and all \(p\geq0\). This assumption is satisfied in particular if \(\mu\) is \(K\)-anti-dominant and sufficiently far from the walls of the Weyl chamber \(-C_K\). They prove in Theorem 7.2 that the tableau \(A\) associated to \(V^\mu\) is \(H^d(Z,N^*_{Z/D}\otimes L_\mu)\) and is involutive. In the same theorem, they also identify the Spencer sequence associated with this tableau and describe the characteristic variety \(\Xi_A\) and the characteristic sheaf. The characteristic module \(M_{\mu,A}\) is also defined in terms of a minimal free resolution. The characteristic sheaf \({\mathcal M}_{\mu,A}\) is then defined in Section 8 as the localisation of \(M_{\mu,A}\). This sheaf has support \(\Xi_A\) and is analysed further in Section 8. Note in particular that \(\Xi_A\) depends only on the Weyl chamber for which \(\mu\) is anti-dominant, while \(M_{\mu,A}\) and \({\mathcal M}_{\mu,A}\) depend on \(\mu\) itself and encode more information than \(\Xi_A\). Section 8 also contains examples which illustrate the geometry behind the construction and the variety of phenomena that arise.
Finally, there is an appendix in which the authors discuss, without formal proofs, the definition of higher symbol maps and characteristic varieties when the vanishing assumption of Sections 7 and 8 does not hold. representations of Lie groups; Harish-Chandra module; PDE; homogeneous space; spectral sequence; correspondence space; characteristic variety Homogeneous spaces and generalizations, Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Semisimple Lie groups and their representations, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Homogeneous complex manifolds, Representation theory for linear algebraic groups, Overdetermined systems of PDEs with variable coefficients, Holomorphic fiber spaces, Differential geometry of homogeneous manifolds, Spectral sequences in algebraic topology On the differential equations satisfied by certain Harish-Chandra modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By using the Milnor ring structure of the cohomology of Grassmannian, we prove a vanishing theorem of Witten genus for generalized string complete intersections in products of Grassmannian. residue; Grassmannian; Witten genus; theta function 32. J. Zhou and X. Zhuang, Witten genus of generalized complete intersections in products of Grassmannians, Int. J. Math.25(10) (2014), Article ID: 1450095, 25pp. [Abstract] genRefLink(128, 'S0129167X16500762BIB032', '000346174500004'); Elliptic genera, Theta functions and abelian varieties, Grassmannians, Schubert varieties, flag manifolds, Complete intersections Witten genus of generalized complete intersections in products of Grassmannians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Affine Schubert calculus is a subject that ties together combinatorics, algebraic geometry and representation theory. Its modern development is motivated by the relation between k-Schur functions and the (co)homology of the affine Grassmannian of \(SL(n)\). The \(k\)-Schur functions were introduced by Lapointe, Lascoux, Morse in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory.
In this paper, the author prove the affine Pieri rule for the cohomology of the affine flag variety conjectured in [\textit{T. Lam} et al., Math. Z. 264, No. 4, 765--811 (2010; Zbl 1230.05279)]. He introduces the cap operators acting on the affine nilCoxeter ring \(A_0\) by investigating the work of Kostant and Kumar and show that the cap operators for Pieri elements are the same as the Pieri operators sing strong strips. The affine Pieri rule gives us a geometric interpretation of the skew strong Schur functions as an affine Grassmannian part of the cap product of the Schubert classes in (co)homology of the affine flag variety. Then he describes these two operators. affine flag variety; k-Schur function; nilCoxeter algebra; Pieri rule; strong Schur function Lee, Seung Jin, Pieri rule for the affine flag variety, Adv. Math., 304, 266-284, (2017) Homogeneous spaces and generalizations, Classical problems, Schubert calculus Pieri rule for the affine flag variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article investigates a conjecture of Beilinson and Bloch which relates the rank of the Griffiths group of a smooth projective variety over a number field to the order of vanishing of an L-function at the center of the critical strip. Numerical data is gathered for 1-cycles on \(E^3\) for 76 different elliptic curves \(E/\mathbb{Q}\). The empirical order of vanishing of the L-function is at least as large as a lower bound on the rank of the Griffiths group coming from the existence of certain genus three curves on \(E^3\). For 11 of the cases considered, the rank of \(\text{Griff}^2(E^3_{\mathbb{Q}})\) is shown to be at least 2. conjecture of Beilinson and Bloch; rank of the Griffiths group; smooth projective variety over a number field; order of vanishing of an L-function; elliptic curves J. Buhler, C. Schoen, and J. Top, ''Cycles, \(L\)-functions and triple products of elliptic curves,'' J. reine angew. Math., vol. 492, pp. 93-133, 1997. \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over global fields, Algebraic cycles, Global ground fields in algebraic geometry Cycles, \(L\)-functions and triple products of elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Using the concept of characteristic polyhedron [\textit{H. Hironaka}, J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)], the authors continue to develop a constructive approach to the resolution of singularities in three-dimensional space in a more general context similar to [\textit{V. Cossart} and \textit{O. Piltant}, Math. Ann. 361, No. 1--2, 157--167 (2015; Zbl 1308.14008)]. The key idea is to determine the Hironaka polyhedron without passing to the completion of the local ring of a singularity. Thus, they prove that this is possible in a number of special situations, including the cases of local Henselian \(G\)-rings and singularities whose defining ideals satisfy certain numerical conditions on their standard bases. singularities; embedded resolutions; polyhedra; Hironaka's characteristic polyhedron; excellent rings; strong normalization; Hilbert-Samuel function; Hironaka schemes; standard bases Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Characteristic polyhedra of singularities without completion. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this work, the authors revisit the determination of the boundary map coefficients for the cellular homology of real flag manifolds, a problem equivalent to finding the incidence coefficients of the differential map for the cohomology. They prove a new formula for these coefficients with respect to the height of certain roots.
A generalized flag manifold \(\mathbb F \) is a homogeneous space \(G/P\), where \(G\) is a real noncompact semisimple Lie group and \(P\) is a parabolic subgroup. It admits a cellular decomposition called Bruhat decomposition, where the cells are the Schubert cells and parametrized by the Weyl group \(W\). There is the Bruhat Chevalley order on elements of the Weyl group. In this case, there is a root \(\beta\) such that \(w = s_{\beta}\cdot w'\). In both [\textit{R. R. Kocherlakota}, Adv. Math. 110, No. 1, 1--46 (1995; Zbl 0832.22020)] and [\textit{L. Rabelo} and \textit{L. A. B. San Martin}, Indag. Math., New Ser. 30, No. 5, 745--772 (2019; Zbl 1426.57052)], the authors summarized how to compute the coefficient \(c(w,w')\).
The papers [\textit{L. Rabelo}, Adv. Geom. 16, No. 3, 361--379 (2016; Zbl 1414.57018)] and [\textit{J. Lambert} and \textit{L. Rabelo}, Australas. J. Comb. 75, Part 1, 73--95 (2019; Zbl 1429.05005)] apply this procedure in the context of the Isotropic Grassmannians and the results obtained (for instance, see [\textit{J. Lambert} and \textit{L. Rabelo}, ``Integral homology of real isotropic and odd orthogonal Grassmannians'', Preprint, \url{arXiv:1604.02177}, to appear in Osaka J. Math.], Theorem 3.12) suggest a formula for the coefficients in terms of the height of some root.
Overall they prove a new formula for the cellular homology coefficients of real flag manifolds in terms of the height of certain roots. For real flag manifolds of type \(A\), they get simple expressions for the coefficients that allow us to compute the first and second integral homology groups exhibiting their generators. real flag manifolds; symmetric group; root systems; Schubert cells; homology; height of roots; boundary coefficients Homology and cohomology of homogeneous spaces of Lie groups, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds, Root systems A correspondence between boundary coefficients of real flag manifolds and height of roots | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is well known that a Schur function is the `limit' of a sequence of Schur polynomials in an increasing number of variables, and that Schubert polynomials generalize Schur polynomials. We show that the set of Schubert polynomials can be organized into sequences, whose `limits' we call Schubert functions. A graded version of these Schubert functions can be computed effectively by the application of mixed shift/multiplication operators to the sequence of variables \(x=(x_1,x_2,x_3,\dots)\). This generalizes the Baxter operator approach to graded Schur functions of G. P. Thomas, and allows the easy introduction of skew Schubert polynomials and functions. Since the computation of these operator formulas relies basically on the knowledge of the set of reduced words of permutations, it seems natural that in turn the number of reduced words of a permutation can be determined with the help of Schubert functions: we describe new algebraic formulas and a combinatorial procedure, which allow the effective determination of the number of reduced words for an arbitrary permutation in terms of Schubert polynomials. Schur function; Schubert polynomials; Schur polynomials; Schubert functions; Baxter operator; reduced words of permutations Winkel, R.: Schubert functions and the number of reduced words of permutations, Sém. lothar. Combin. 39, 1-28 (1997) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Schubert functions and the number of reduced words of permutations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper presents a construction of the category of perverse sheaves which are constructible relative to a given stratification which is ''elementary'' in the sense that derived categories do not appear explicitly. The construction is inductive: if S is a closed stratum in X the categories for X and X-S are related by the following construction. One has two categories A and B, two functors \(F,G: A\to B\) and a natural transformation \(T: F\to G;\) and forms the category of factorizations FA\(\to B\to GA\) of maps TA. The theoretical construction of this data involves operations with derived direct and inverse image functors of categories of sheaves. However, these can be made more concrete, assuming all strata even dimensional and using the middle perversity.
Calculations centre round a ''perverse link'' which can be constructed using the second barycentric subdivision of a triangulation of the stratified link or, in the complex analytic case, more directly: the definition is by conditions on its intersection homology. If S is 2- connected this data suffices for the construction; in general consideration of the ''perverse link bundle'' is required.
The authors apply their results to give a topological proof of a theorem of \textit{M. Kashiwara} and \textit{T. Kawai} [Publ. Res. Inst. Math. Sci. 17, 813-979 (1981; Zbl 0505.58033)] giving (in the complex case) a simple criterion for the extension to be trivial. They also give a number of examples; in the simplest, the complex line stratified by the origin and its complement, the sheaf is characterized by the monodromy. In the other examples also the authors show the category is equivalent to the category of representations of some fixed quiver, and conjecture that this is always so for certain natural stratifications of the space of Borel subgroups of a reductive complex algebraic group. representations of quiver; construction of the category of perverse sheaves; closed stratum; middle perversity; barycentric subdivision; intersection homology; perverse link bundle; monodromy; Borel subgroups; reductive complex algebraic group MacPherson, R.; Vilonen, K., Elementary construction of perverse sheaves, \textit{Invent. Math.}, 84, 403-435, (1986) Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Finite rings and finite-dimensional associative algebras, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), de Rham cohomology and algebraic geometry, Differential complexes, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Elementary construction of perverse sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study algebraic varieties parametrized by topological spaces and enlarge the domain of Lawson homology and morphic cohomology to this category. We prove a Lawson suspension theorem and a splitting theorem. A version of the Friedlander-Lawson moving lemma is obtained to prove a duality theorem between Lawson homology and morphic cohomology for smooth semi-topological projective varieties. \(K\)-groups for semi-topological projective varieties and Chern classes are also constructed. Lawson homology; morphic cohomology; semi-topological varieties; Friedlander-Lawson moving lemma; Hilbert scheme; Chow varieties Algebraic cycles, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Semi-topological cycle theory I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{E. Witten} [Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005)] interpreted the Jones invariants of links in \(S^3\) (such as the Jones polynomial) in terms of his topological quantum field theory using Chern-Simons theory (see \textit{K. Marathe} [in: The mathematics of knots. Theory and application. Banagl, Markus (ed.) et al., Berlin: Springer. Contributions in Mathematical and Computational Sciences 1, 199--256 (2011; Zbl 1221.57023)] for a recent survey). This theory depends on a rank \(N\) and level \(k\), the quantum Hilbert space being identified with level \(k\) highest weight representations of the Lie algebra corresponding to \(\text{SU}(N)\) and acted on by the group \(\text{SL}(2,\mathbb Z)\) via matrices \(S\) and \(T\). Recently \textit{M. Aganagic} and \textit{Sh. Shakirov} proposed a refinement of the \(\text{SU}(N)\) Chern-Simons theory for links in 3-manifolds having \(S^1\) symmetry [String-Math 2011, Proc. Symp. Pure Math. 85, Amer. Math. Soc., Providence RI 2012, 3--31 (2012)] in which the matrices \(S\) and \(T\) are replaced by matrices used by \textit{I. Cherednik} [Invent. Math. 122, No. 1, 119--145 (1995; Zbl 0854.22021)] and \textit{A. A. Kirillov, jun.} [J. Am. Math. Soc. 9, No. 4, 1135--1169 (1996; Zbl 0861.05065)]. In the refined theory, the Hilbert space is identified with the MacDonald polynomials of type \(\text{SU}(N)\) with parameters \(q,t\) satsifying \(q^k t^N = 1\).
In the paper under review, the author computes the limit of the matrix \(S\) as \(N \to \infty\) for the refined theory. Starting with the explicit form of \(S\) given by \textit{M. Aganagic} and \textit{Sh. Shakirov} [``Knot homology from refined Chern-Simons theory'', Preprint, 2011, \url{arXiv:1105.5117}], he replaces \(t^N\) by a variable \(u\) to obtain a stable version expressed in terms of the modified MacDonald polynomials of \textit{A. Garsia} et al. [Sémin. Lothar. Comb. 42, B42m, 45 p. (1999; Zbl 0920.05071)]. To compute the kernel function, he starts with the Cherednik-MacDonald-Mehta identity in the form used by Garsia, Haiman and Tesler [loc. cit.] and using the relation between MacDonald polynomials and Hilbert schemes of \(n\) points in \(X=\mathbb C^2\) due to \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001), Invent. Math. 149, No. 2, 371--407 (2002; Zbl 1053.14005)] he expresses two of the terms as power series in \(u\): the coefficient of \(u^n\) in each case is an equivariant Euler characteristics of certain sheaves on the Hilbert scheme \(X^{[n]}\) for the group action of \(\mathbb C^* \times \mathbb C^*\) - in one case the structure sheaf \({\mathcal O}_{X^{[n]}}\) and in the other case the tensor product of arbitrary Schur functors \(s_{\lambda}\) and \(s_{\mu}\) applied to a universal sheaf. Hilbert scheme of points in the plane; Chern-Simons theory Hiraku Nakajima, Refined Chern-Simons theory and Hilbert schemes of points on the plane, Perspectives in representation theory, Contemp. Math., vol. 610, Amer. Math. Soc., Providence, RI, 2014, pp. 305 -- 331. Parametrization (Chow and Hilbert schemes), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Refined Chern-Simons theory and Hilbert schemes of points on the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective variety. A line bundle \(L\) on \(X\) is said to be \(3\)-very ample if the restriction map \(H^0(L) \to H^0(L|Z)\) is surjective for every degree \(4\) zero-dimensional subscheme \(Z\subset X\). The main result proved here is that if \(L\) is \(3\)-very ample and it has another technical assumption, then the secant variety \(\Sigma (X,L)\) of \(X\) of the image of \(X\) by the complete linear system \(|L|\) is normal. This technical assumption is satified if \(L\cong \omega _X\otimes A^{\otimes 2(n+1)}\otimes B\), \(n:= \dim (X)\), with \(A\) very ample and \(B\) nef. If \(X\) is a smooth curve of genus \(g\), it is sufficient to assume that \(\deg (L)\geq 2g+3\). She also proves the normality of the secant variety of the canonical embedding of all smooth curves with Clifford index \(\geq 3\). These are very nice results and they also fill a gap in the published proofs by other authors.
Among the papers used for the proofs or generalized by the present one or whose quotations are now fixed I mention: [\textit{A. Bertram}, J. Differ. Geom. 35, No. 2, 429--469 (1992; Zbl 0787.14014)], [\textit{J. Sidman} and \textit{P. Vermeire}, Algebra Number Theory 3, No. 4, 445--465 (2009; Zbl 1169.13304); Abel Symposia 6, 155--174 (2011; Zbl 1251.14043)], [\textit{P. Vermeire}, Compos. Math. 125, No. 3, 263--282 (2001; Zbl 1056.14016); J. Algebra 319, No. 3, 1264--1270 (2008; Zbl 1132.14027); Proc. Am. Math. Soc. 140, No. 8, 2639--2646 (2012; Zbl 1279.14068)]. secant variety; normality of secant variety; \(k\)-very ample line bundle; Hilbert scheme; positivity; vector bundle Ullery, B.: On the normality of secant varieties. Adv. Math. \textbf{288}, 631-647 (2016). arXiv:1408.0865v2 Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes), Plane and space curves, Questions of classical algebraic geometry On the normality of secant varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Es sei \(K\) ein Körper, der über seinem Primkörper endlich erzeugbar ist, und \(V\) das Skelett einer abstrakten arithmetischen Varietät zu \(K\), d.h. die Gesamtheit der lokalen Ringe dieser Varietät (diese gehen durch Quotientenringbildung aus absolut endlich erzeugbaren Teilringen von \(K\) hervor). Kähler schlug eine Definition einer Zetafunktion von \(V\) vor, die sich folgendermaßen schreiben läßt:
(1) \(\zeta(V|s)=\prod_{\mathfrak P}\zeta_{\mathfrak P}(s)\), wobei das Produkt über alle \(O\)-dimensionalen lokalen Ringe \(S\) von \(V\) (bzw. ihre maximalen Primideale \(\mathfrak P\)) läuft und
(2) \(\zeta_{\mathfrak P}(s)=\sum_{\mathfrak g}N(\mathfrak g)^{-s}\) ist (\(\mathfrak g\) jeweils alle Ideale mit endlicher Restklassenanzahl \(N(\mathfrak g)\) von \(S\)). Der Ausdruck (1) läßt sich leicht in eine formale Dirichlet-Reihe umschreiben; ebenso kann man in Verallgemeinerung hiervon Zetafunktionen zu ``Gebilden'' auf \(V\) einführen.
Verf. betrachtet nun den Fall eines Körpers \(K\) der Stufe \(2\) (\(K=\) algebraischer Funktionenkörper von zwei Veränderlichen über einem Galoisfeld oder \(K =\) algebraischer Funktionenkörper von einer Veränderlichen über einem Zahlkörper) unter der Voraussetzung, daß alle \(\mathfrak P\) regulär sind, und bestimmt analytische Eigenschaften von \(\zeta(V|s)\). Hierzu zeigt Verf. im ersten Teil für die Anzahl \(g_\lambda\) der Ideale \(\mathfrak a\) von \(S\) mit fester endlicher Restklassenanzahl \(N(\mathfrak a) = N(\mathfrak p)^{\lambda}\): Die Anzahl \(g_\lambda\) ist ein Polynom vom genauen Grade \(\lambda-1\) in \(N(\mathfrak p)\) mit positiven ganzen Koeffizienten, für welche eine Rekursionsformel hergeleitet wird. Als Hilfüberlegung dient hierbei: Es sei das maximale Ideal \(\mathfrak p\) von \(S\) durch die zwei Elemente \(u,v\) erzeugt, \(b\) sei das von \(v\) in \(S\) erzeugte Ideal, und für zwei Ideale \(\mathfrak a, \mathfrak b\) von \(S\) mit \(\mathfrak a\supset\mathfrak b\) bezeichne \((\frac{\mathfrak b}{\mathfrak a})\) die Länge einer Kompositionsreihe von \(\mathfrak a\) nach \(\mathfrak b\). Sind dann \(\nu_0,\nu_1,\dots,\nu_n,\nu_{n+1},\dots\) ganze Zahlen mit \(\nu_0\geq\nu_1\geq\dots\geq\nu_n>0=\nu_{n+1}=\nu_{n+2}=\dots,\) so ist die Anzahl \(f(\nu_0,\nu_1,\dots)\) der Ideale \(\mathfrak a\) von \(S\) mit
\[
((\mathfrak a+\nu^{i+1})/(\mathfrak a+\nu^i))=\nu_i,~ i=0,1,\dots,
\]
gegeben durch
\[
f(\nu_0,\nu_1,\dots)=N(\mathfrak P)^{\nu_1+\nu_2+\dots+\nu_n}.
\]
Hieraus folgt, daß (2) für \(\Re(s) > 1\) absolut konvergiert zum Werte
\[
(3) \zeta_{\mathfrak P}(s)=\prod_{\rho=0}^{\infty}(1-N(\mathfrak P)^{\rho-(\rho+1)s})^{-1}.
\]
Mit diesem Ausdruck zeigt Verf. durch eine algebraische Zurückführung auf den rationalen Fall (Stufe 1) das folgende Hauptergebnis: Im vorliegenden Fall konvergiert (1) für \(\Re(s) > 2\) absolut gegen eine dort analytische Funktion; das gleiche gilt für die Zetafunktion der ``Gebilde'' auf \(V\).
Für kritische Beispiele zu den analytischen Eigenschaften von \(\zeta(V|s)\) vgl. auch Ref. [Durch Produktdarstellungen erklärte Zetafunktionen. Sammelband Leonhard Euler, Dtsch. Akad. Wiss. Berlin 246--255 (1959; Zbl 0121.04602)]; die unangenehmen Eigenschaften liegen im Wesentlichen daran, daß der lokale Beitrag \(\zeta_{\mathfrak P}(s)\) aus (3) wesentlich komplizierter gebildet ist als der entsprechende bei A. Weil und anderen.
Verf. benutzt die vom üblichen abweichende Kählersche Bezeichnungsweise. number fields; function fields G. Lustig, Über die Zetafunktion einer arithmetischen Mannigfaltigkeit. Math. Nachr.14, 309--330 (1956). Arithmetic varieties and schemes; Arakelov theory; heights, Zeta and \(L\)-functions: analytic theory On the zeta function of an arithmetic manifold. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In previous work the author has developed a theory of operator vessels associated with operators \(A_1,A_2,\dots, A_n,B_1,B_2,\dots,B_n\) on a Banach space such that the differences \(A_j-B_j\) are small (e.g., of finite rank). A special kind of vessel can be constructed when the operators \(A_j,B_j\) are obtained as rational functions of operators \(A, B\) (i.e., \(A_j=r_j(A)\) and \(B_j=r_j(B))\) with rank one difference \(A-B\). The author makes a general conjecture about sufficient conditions under which a given vessel can be identified with one of the special kind. He proves this conjecture for \(n=2\). operator colligations; operator vessels; Livsic characteristic function; invariant subspaces; rational functions of operators Kravitsky, N, No article title, Integral Equations Operator Theory, 26, 60, (1996) Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc., Rational and birational maps, Equations and inequalities involving linear operators, with vector unknowns, Transformers, preservers (linear operators on spaces of linear operators), Functional calculus for linear operators Rational operator functions and Bezoutian operator vessels | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0614.00006.]
Let \(H(d,g)_ S\) be the open subscheme of the Hilbert scheme of curves of degree \(d\) and arithmetic genus g in \({\mathbb{P}}^ 3\) parametrizing smooth irreducible curves. The first author who pointed out the existence of irreducible non reduced components of \(H(d,g)_ S\), was \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.131)] who found a non reduced component of \(H(14,24)_ S\), the general curve of which lies on a smooth cubic surface in \({\mathbb{P}}^ 3\). Mumford's example has been widely generalized by the author of the present paper in his thesis (``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space'', Preprint no. 5-1981, Univ. Oslo). Among other things it turns out from his analysis that if \(W\subseteq H(d,g)_ S\) is a closed irreducible subset whose general point corresponds to a curve C lying on a smooth cubic surface, W is maximal under this condition and \(d>9\), then W irreducible, non reduced component of \(H(d,g)_ S\) yields \(g\geq 3d-18\) and \(H^ 1({\mathcal J}_ C(3))\neq 0\) (the latter inequality implying that \(g\leq (d^ 2-4)/8.\)
The author conjectures that these necessary conditions are also sufficient for W to be a non reduced component of \(H(d,g)_ S\), and he proves this conjecture in the ranges \(7+(d-2)^ 2/8<g\leq (d^ 2-4)/8\), \(d\geq 18\) and \(-1+(d^ 2-4)/8<g\leq (d^ 2-4)/8\), 17\(\geq d\geq 14\). The proof consists in an interesting analysis of the tangent and obstruction space to the so called Hilbert-flag scheme (parametrizing pairs (curve, surface), the first contained in the latter) in particular for curves lying on surfaces of degree \(s\leq 4.\) space curves; Hilbert scheme; degree; arithmetic genus; obstruction space; Hilbert-flag scheme J. O. Kleppe, Nonreduced components of the Hilbert scheme of smooth space curves. In Space curves (Rocca di Papa, 1985), volume 1266 of Lecture Notes in Math. (Springer, Berlin, 1987), pp. 181-207. Zbl0631.14022 MR908714 Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry Non-reduced components of the Hilbert scheme of smooth space curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers an algebraic extension \(K\) of \(\mathbb{Q}_p\) and denotes by \({\mathcal O}\) its ring of integers. He obtains a converse of the statement that if \(f(x)\) is a noninvertible endomorphism of a formal group, then \(f(x)\) commutes with an invertible series and \(\overline {\mathcal O} [[f^n(x)]] \overline {\mathcal O} [[x]]\) is Galois for all \(n\). He uses the Weierstraß preparation theorem, Newton polygons and techniques of \textit{J. Lubin} on nonarchimedean dynamical systems [Compos. Math. 94, 321-346 (1994; Zbl 0843.58111)]. In the main result of the paper he proves that if \(f(x) \in {\mathcal O} [[x]]\) is a noninvertible stable series such that the extension of the ring \(\overline {\mathcal O} [[x]] \supset \overline {\mathcal O} [[f(x)]]\) is Galois with Galois group \(\Gamma\), then \(f(x)\) is an endomorphism of a formal group over \({\mathcal O}\). \(p\)-adic power series; \(p\)-adic dynamical systems; noninvertible endomorphism of a formal group Li, Hua-Chieh, When is a \(p\)-adic power series an endomorphism of a formal group?, Proc. Amer. Math. Soc., 124, 8, 2325-2329, (1996) Class field theory; \(p\)-adic formal groups, Formal groups, \(p\)-divisible groups When is a \(p\)-adic power series an endomorphism of a formal group? | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Hilbert in his famous paper in 1890 used the projective resolution of a graded module \(M\) over \(k[X_ 1, \dots, X_ n]\) to create its Hilbert function. The authors, in this paper, consider the converse problem. That is, given a Hilbert function of a graded module \(M\), they consider all possible projective resolutions with this Hilbert function. They consider only cyclic modules. They are successful only in the case of rings of dimension two. If \(R = k[x,y]\) is a two-dimensional graded regular ring and if \(R/I\) is a graded module of finite length with minimal resolution, they find, in theorem 1, some conditions on degrees of generators and relations. The converse of this is dealt with in theorem 2 as a consequence of which we can get, for a given Hilbert series of a cyclic module, resolutions with the smallest or biggest possible graded Betti numbers. It is illustrated with a concrete example. In the higher dimensional case the situation is not so satisfactory. If the dimension of \(R = k[x_ 1, \dots, x_ n]\) is at least three, there exist a Hilbert series for a cyclic module of finite length and two incomparable smallest sets of graded Betti numbers for that series. Betti number; Hilbert polynomial; projective resolution; graded module; Hilbert function; Hilbert series Hara Charalambous and E. Graham Evans Jr., Resolutions with a given Hilbert function, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 19 -- 26. Resolutions; derived functors (category-theoretic aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes), Deformations and infinitesimal methods in commutative ring theory Resolutions with a given Hilbert function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper focuses on families of Artinian or one-dimensional quotients of a polynomial ring \(R\). Let \(H\) be a Hilbert function and let GradAlg\(^H (R)\) be the scheme parametrizing all graded quotients of \(R\) with Hilbert function \(H\). Let \(B \rightarrow A\) be a graded surjection of quotients of \(R\), with Hilbert functions \(H_B\) and \(H_A\) respectively. If \(\dim A = 0\) or 1 and making some additional assumptions on both \(A\) and \(B\), the author gives close connections between GradAlg\(^{H_B}(R)\) and GradAlg\(^{H_A}(R)\). These connections involve for instance smoothness and dimension of these parameter schemes. In a more general setting he describes the dual of the tangent and obstruction space of graded deformations. He then applies this machinery to the case of level algebras of Cohen-Macaulay type 2 (this is a natural first case after the Gorenstein algebras). As a result, he proves a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg\(^H(R)\) when \(H = (1,3,6,10,14,10,6,2)\), such that the general elements of these components are level. The work is very technical, but many examples are given to illustrate the methods. Similar parameter schemes have been studied in the past, but generally those papers have considered the reduced scheme structure, while GradAlg\(^H (R)\) may be non-reduced. parametrization; Artinian algebra; level algebra; Gorenstein algebra; licci; Hilbert scheme; duality; algebra (co)homology; canonical module; normal module Kleppe, J.O., Families of Artinian and one-dimensional algebras, J. algebra, 311, 665-701, (2007) Parametrization (Chow and Hilbert schemes), Commutative Artinian rings and modules, finite-dimensional algebras, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Deformations and infinitesimal methods in commutative ring theory, Linkage, complete intersections and determinantal ideals Families of Artinian and one-dimensional algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{A. S. Gorsky} et al. [Nucl. Phys. B 527, 690-716 (1998; Zbl 0951.37020)]\ presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the \(\text{SU}(N + 1)\) \(N=2\) SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the \(A^{(2)}_{2N}\) affine Toda system. Our construction, too, uses fractional powers of the superpotential \(W(x)\) that characterizes the curve. We also consider the \(u\)-plane integral of topologically twisted theories on four-dimensional manifolds \(X\) with \(b_2^+(X)=1\) in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind. spectral curve; affine Toda system; fractional powers; superpotential K. Takasaki, \textit{Whitham deformations of Seiberg-Witten curves for classical gauge groups}, \textit{Int. J. Mod. Phys.}\textbf{A 15} (2000) 3635 [hep-th/9901120] [INSPIRE]. Groups and algebras in quantum theory and relations with integrable systems, Relationships between algebraic curves and integrable systems, Topological field theories in quantum mechanics, Supersymmetric field theories in quantum mechanics, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Yang-Mills and other gauge theories in quantum field theory Whitham deformations of Seiberg-Witten curves for classical gauge groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study category-theoretic properties of monomorphisms in categories of log schemes. This study allows one to give a purely category-theoretic reconstruction of the log scheme that gave rise to the category under consideration. We also obtain analogous results for categories of schemes of locally finite type over the ring of rational integers that are equipped with ``Archimedean structures''. Such reconstructions were discussed in two previous papers by the author [Adv. Math. 188, No. 1, 222--246 (2004; Zbl 1073.14002); J. Math. Kyoto Univ. 44, No. 4, 891--909 (2004; Zbl 1087.14505)], but these reconstructions contained some errors, which were pointed out to the author by C. Nakayama and Y. Hoshi. These errors revolve around certain elementary combinatorial aspects of fan decompositions of two-dimensional rational polyhedral cones -- i.e., of the sort that occur in the classical theory of toric varieties -- and may be repaired by applying the theory developed in the present paper. log scheme; scheme; category; anabelian geometry; Archimedean structure Schemes and morphisms, Arithmetic varieties and schemes; Arakelov theory; heights Monomorphisms in categories of log schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\sigma =\sigma_ K=\sigma_{{\mathbb{Q}}(\sqrt{- 3})}={\mathbb{Z}}+{\mathbb{Z}}\omega\), \(\omega =e^{\pi i/3}\), be the ring of integral Eisenstein numbers and \(\Gamma =U((2,1),\sigma)\) be a lattice in U((2,1),\({\mathbb{C}})\). Denote by \(\Gamma '\) the congruence subgroup of \(\Gamma\) with respect to the prime ideal \((1-\omega)\sigma\). Then \(\Gamma/\Gamma' \cong \sigma_ 4\) the symmetric group of 4 letters.
Let B be the complex 2-ball, let \((B/\Gamma)\) be the Baily-Borel-Satake compactification of the open algebraic surface \(B/\Gamma\) and let \(K(B/\Gamma)\) be the field of K-Picard modular functions. A singular module on B is an isolated fixed point of (elements of) the group \(GU((2,1),K)=\{\gamma \in GL_ 3({\mathbb{C}})|\) \(<\gamma a,\gamma b>=c_{\gamma}<a,b>\) for all \(a,b\in {\mathbb{C}}^ 3\}\) where \(<.,.>\) denotes the Hermitian scalar product on \({\mathbb{C}}^ 3\) with signature (2,1).
The main result of the paper is that for any singular module \(\sigma\) and any K-Picard modular function \({\mathcal F}\), \({\mathcal F}(\sigma)\) is an algebraic number. arithmetic points; algebraicity; integral Eisenstein numbers; Picard modular function Holzapfel, R.-P. : An arithmetic uniformization for arithmetic points of the plane by singular moduli , J. Ramanujan Math. Soc. 3(1), (1988), S.35-62. Theta series; Weil representation; theta correspondences, Global ground fields in algebraic geometry, Families, moduli, classification: algebraic theory An arithmetic uniformization for arithmetic points of the plane by singular moduli | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems On the basis of a software implementation of Kummer based HECC over \(\mathbb {F}_p\) presented in 2016, we propose new hardware architectures. Our main objectives are: definition of architecture parameters (type, size and number of units for arithmetic operations, memory and internal communications); architecture style optimization to exploit internal parallelism. Several architectures have been designed and implemented on FPGAs for scalar multiplication acceleration in embedded systems. Our results show significant area reduction for similar computation time than best state of the art hardware implementations of curve based solutions. hyper-elliptic curve cryptography; hardware implementation; architecture exploration; embedded systems Cryptography, Applications to coding theory and cryptography of arithmetic geometry Architecture level optimizations for Kummer based HECC on FPGAs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb{K}\) be a field of characteristic zero, \(R = \mathbb{K}[x_{1},\dots, x_{n}]\) a polynomial ring over \(\mathbb{K}\) and \(N\) a set of differential operators in the corresponding Weyl algebra \(W_{R}\), that is, in the ring of differential operators over \(R\) generated by the partial derivations \(\partial_{i} = \partial/\partial x_{i}\) (\(1\leq i\leq n\)). If \(I\) is an ideal of \(R\), then a set \(N\subseteq W_{R}\) is called a set of Noetherian operators for \(I\) if \(f\in I\) if and only if \(D\bullet f\in\sqrt{I}\) for any \(D\in N\). (\(\bullet\) denotes the action of \(D\) on \(f\) (\(x_{i}\bullet f = x_{i}f\), \(\partial_{i}\bullet f = \partial f/\partial x_{i}\)) and \(\sqrt{I}\) denotes the radical of \(I\).) Symbolic algorithms to compute Noetherian operators were developed and implemented in [\textit{A. Damiano} et al., Exp. Math. 16, No. 1, 41--53 (2007; Zbl 1136.13014); \textit{Y. Cid-Ruiz} et al., ``Primary ideals and their differential equations'', Preprint, \url{arXiv: 2001.04700}].
The paper under review addresses the problem of representing a (primary) polynomial ideal via a dual set of differential operators. In this connection, the authors present new algorithms to compute a set of Noetherian operators representing a primary ideal, as well as theoretical results leading up to them. One of the algorithms is a symbolic one; it reduces the problem to a problem of linear algebra in the spirit of the approach explored in [\textit{F. S. Macaulay}, The algebraic theory of modular systems. Cambridge: University Press (1916; JFM 46.0167.01)]. Another developed algorithm is a numerical one, it may solve problems that are out of reach for purely symbolic techniques (the authors illustrate this by a corresponding example). Given an ideal with no embedded components, this numerical algorithm combined with numerical irreducible decomposition leads to numerical primary decomposition, that is, a numerical description of all components of the ideal, which e.g. enables a probabilistic membership test. Noetherian operators; inverse systems; primary decomposition Computational aspects of higher-dimensional varieties, Numerical methods for differential-algebraic equations, Modules of differentials, Numerical interpolation Noetherian operators and primary decomposition | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a smooth irreducible algebraic curve of genus at least two and \(\Aut (C)\) be its group of automorphisms. The authors describe possible groups of \(\Aut (C)\) in the case \(C\) admits only a finite number of base-point-free linear systems \(g^1_p\) with \(p\)-prime, and apply this description to some more special classes of curves. automorphisms of algebraic curve; linear systems González, V.; Rodríguez, R.: On automorphisms of curves and linear series, Aportaciones mat. Notas investigación 5, 101-105 (1992) Automorphisms of curves, Curves in algebraic geometry, Divisors, linear systems, invertible sheaves On automorphisms of curves and linear series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The problem of compactifying the (generalized) Jacobian of a singular curve has been studied since \textit{J. Igusa} [Am. J. Math. 78, 171--199, 745--760 (1956; Zbl 0074.15803)]. He constructed a compactification of the Jacobian of a nodal and irreducible curve \(X\) as limit of the Jacobians of smooth curves approaching \(X.\) Igusa also showed that his compactification does not depend on the considered family of smooth curves. When the curve \(X\) is reducible and nodal, \textit{T. Oda} and \textit{C. S. Seshadri} [Trans. Am. Math. Soc. 253, 1--90 (1979; Zbl 0418.14019)] produced a family of compactified Jacobians \(\text{Jac}_{\phi}\) parameterized by an element \(\phi\) of a real vector space. \textit{C. S. Seshadri} [``Fibrés vectoriels sur les courbes algébriques'', Astérisque 96 (1982; Zbl 0517.14008)] dealt with the general case of a reduced curve considering sheaves of higher rank as well. \textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)] showed how to compactify the relative Jacobian over the moduli of stable curves and described the boundary points of the compactified Jacobian of a stable curve \(X\) as invertible sheaves on certain Deligne-Mumford semistable curves that have \(X\) as a stable model. Most of the above papers are devoted to the construction of the compactified Jacobian of a curve, not to describe it.
In the paper under review, the author gives an explicit description of the structure of these Simpson schemes, \(\text{Jac}^{d}(X)_{s},\) and \(\overline{\text{Jac}^{d}(X)}\) of any degree \(d,\) for \(X\) a polarized curve of the following types: tree-like curves and all reduced and reducible curves that can appear as singular fibers of an elliptic fibration. moduli of stable curves; Deligne-Mumford semistable curves; Simpson scheme López~Martín, A.C., Simpson Jacobians of reducible curves, J. reine angew. math., 582, 1-39, (2005) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Vector bundles on curves and their moduli, Picard groups, Schemes and morphisms, Families, moduli of curves (analytic) Simpson Jacobians of reducible curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We classify the prelocalizing subcategories of the category of quasi-coherent sheaves on a locally noetherian scheme. In order to give the classification, we introduce the notion of a local filter of subobjects of the structure sheaf. The essential part of the argument is given as results on a Grothendieck category with certain properties. We also classify the localizing subcategories, the closed subcategories, and the bilocalizing subcategories in terms of filters. locally noetherian scheme; prelocalizing subcategory; localizing subcategory; closed subcategory; local filter Kanda, R., Classification of categorical subspaces of locally Noetherian schemes, Doc. Math., 20, 1403-1465, (2015) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Grothendieck categories, Module categories in associative algebras, Noncommutative algebraic geometry, Structure, classification theorems for modules and ideals in commutative rings Classification of categorical subspaces of locally noetherian schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A projective \([n,k+1]_q\)-system is a set \(X\) of \(n\) points in a projective space over \({\mathbb{F}}_q\) that span a linear subspace of dimension \(k\). The terminology is due to \textit{M. A. Tsfasman} and \textit{S. G. Vlăduţ} [``Algebraic-geometric codes'' (1991; Zbl 0727.94007)] and gives a more invariant way of considering \(q\)-ary linear codes of length \(n\) and dimension \(k\). The authors consider the Hilbert function \(H_X\) of the homogeneous coordinate ring of this set of points, and the first difference function \(\Delta H_X\), which is also called the Castelnuovo function. While the description of the possible Hilbert functions of finite subsets of projective space over an algebraically closed field is known [cf. \textit{A. V. Geramita, P. Maroscia}, and \textit{L. G. Roberts}, J. Lond. Math. Soc., II. Ser. 28, 443-452 (1983; Zbl 0535.13012)], this paper gives a first step toward such a description when the field is finite.
The Castelnuovo-Mumford regularity \(r_X\) is the maximum integer \(\nu\) such that \(\Delta H_X(\nu)\neq 0\). The authors define \(s(n,k,q)\) to be the supremum of the \(r_X\)'s as \(X\) runs through all projective \([n,k+1]_q\)-systems and they define \(s(n,q)\) to be the supremum of the \(s(n,k,q)\)'s for \(1\leq k<n\). Geometrically, \(s(n,q)\) is the least number \(s\) such that for every set of \({\mathbb{F}}_q\)-rational points \(\{P_1,P_2,\dots,P_n\}\) in some projective space over \({\mathbb{F}}_q\), there exists a hypersurface of degree \(s\) containing \(P_1,\dots,P_{n-1}\), but not \(P_n\). The authors give a lower bound for \(s(n,q)\), which gives the exact value when \(q=2\). They also determine a number of other values and estimates for \(s(n,k,q)\) and \(s(n,q)\), and conjecture how \(s(n,q)\) grows asymptotically with \(n\). Applications to coding theory, commutative algebra, and boolean algebra are discussed. projective system; Hilbert function; Castelnuovo-Mumford regularity; Castenuovo function; rational points; coding theory; boolean algebra M. Kreuzer and R. Waldi, On the Castelnuovo-Mumford regularity of a projective system , Comm. Alg. 25 (1997), 2919-2929. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Geometric methods (including applications of algebraic geometry) applied to coding theory, Finite ground fields in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Rational points, Projective techniques in algebraic geometry On the Castelnuovo-Mumford regularity of a projective system | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.